hyperbolic hubbard-stratonovich transformations and bosonisation of granular fermionic systems

Hyperbolic Hubbard-Stratonovichtransformations

andbosonisation of granular fermionic

systems

I n a u g u r a l - D i s s e r t a t i o n

zur

Erlangung des Doktorgrades

der Mathematisch-Naturwissenschaftlichen Fakultat

der Universitat zu Koln

vorgelegt von

Jakob Muller-Hill

aus Koln

2009

Berichterstatter Prof. Dr. Martin Zirnbauer

Prof. Dr. Hans-Peter Nilles

Tag der mundlichen Prufung: 26 Juni 2009

i

Zusammenfassung

Die vorliegende Arbeit besteht aus zwei Teilen. Der erste Teil beschaftigtsich mit hyperbolischen Hubbard-Stratonovich-Transformationen. SolcheTransformationen werden z.B. im Bereich der ungeordneten Elektronensys-teme benotigt, um nichtlineare Sigma-Modelle herzuleiten, die das Niederen-ergieverhalten dieser Systeme beschreiben. Der mathematische Status hy-perbolischer Hubbard-Stratonovich-Transformationen vom Pruisken-Schafer-Typ war lange ungeklart. Kurzlich wurden zwei Spezialfalle, namlich diepseudounitarer und pseudoorthogonaler Symmetrie, bewiesen [10, 11, 12].In dieser Arbeit wird nun der Fall einer allgemeinen (im wesentlichen halb-einfachen) Symmetriegruppe bewiesen. Der Beweis ist anschaulich und zeigtexplizit den Zusammenhang mit Standard-Gauß-Integralen.

Im zweiten Teil wird eine eine neuartige Methode entwickelt, um wech-selwirkende granular fermionische Systeme zu bosonisieren. Die Methodeist nicht mit der bekannten Bosonisierung (1 + 1)-dimensionaler Systemeverwandt, sondern eher im Bereich der koharenten Zustande anzusiedeln.Ein Zugang ist, die Grassmann-Pfadintegraldarstellung einer großkanon-ischen Zustandssumme durch mehrfache Anwendung der Colour-Flavour-Transformation in eine Form zu bringen, welche die Eliminierung der Grass-mannvariablen erlaubt. Das Resulat ist ein Pfadintegral in generalisiertenkoharenten Zustanden mit speziellen Randbedingungen.

ii

Abstract

The present work consists of two parts. The first part deals with hyperbolicHubbard-Stratonovich transformations. Such transformations are used toderive non-linear sigma models that describe the low energy behaviour ofdisordered electron systems. For a long time the mathematical status ofhyperbolic Hubbard-Stratonovich transformations of Pruisken-Schafer typeremained unclear. Only recently the two special cases of pseudounitary andpseudoorthogonal symmetry were proven [10, 11, 12]. In this thesis we provethe transformation for a general (essentially semisimple) symmetry group.The proof is descriptive and shows explicitly the connection to the standardGaussian integrals.

In the second part we develop a novel method to bosonise granularfermionic systems. The method is related to the method of coherent states.In particular it is not based on the well known bosonisation of (1 + 1)-dimensional systems. One approach is to use the colour-flavour transfor-mation to transform the Grassmann path integral representation of a grandcanonical partition function in a way that allows to eliminate the Grassmannvariables. The result is a path integral in generalised coherent states withspecial boundary conditions.

Contents

Introduction v

1 Hyperbolic Hubbard-Stratonovich transformations 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Two dimensional example . . . . . . . . . . . . . . . . . . . . 31.3 General setting and theorem . . . . . . . . . . . . . . . . . . . 91.4 Proof of the theorem . . . . . . . . . . . . . . . . . . . . . . . 111.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Bosonisation of granular fermionic systems 372.1 Granular bosonisation via colour-flavour transformation . . . 372.2 Fock space approach to granular bosonisation . . . . . . . . . 442.3 Contributions of fluctuations . . . . . . . . . . . . . . . . . . 46

Conclusion 57

A Techniques needed in chapter one 59A.1 Basic constructions and useful relations . . . . . . . . . . . . 59A.2 Three additional arguments . . . . . . . . . . . . . . . . . . . 64

B Techniques needed in chapter two 67B.1 Canonical transformations of fermionic Fock space . . . . . . 67B.2 Fermionic Howe pairs and colour-flavour transformation . . . 70

Bibliography 77

iii

iv CONTENTS

Introduction

To obtain an adequate description of a physical system, and to computequantities of interest, it is often necessary to replace the microscopic degreesof freedom of the system by physically more relevant ‘collective’ degrees offreedom. Two prominent methods to introduce collective variables in thefield of many particle physics are Hubbard-Stratonovich transformations andbosonisation. In this work we discuss special variants of both methods. Thefirst part of this work clarifies the mathematical status of a class of hyper-bolic Hubbard-Stratonovich transformations, whereas in the second part anew kind of bosonisation is developed. The focus of this work is rather onmethodology than on applications.

Let us start with a more detailed introduction to the first part of thethesis. First, we explain where hyperbolic Hubbard-Stratonovich transfor-mations are commonly used. A natural area of application of hyperbolicHubbard-Stratonovich transformations are disordered electron systems [1]and their description in the form of non-compact non-linear sigma models.The corresponding formalism was pioneered by Wegner [3], Schafer & Weg-ner [4], and Pruisken & Schafer [5]. Efetov [2] developed the more rigoroussupersymmetry method, which avoids the use of the replica trick, to derive(supersymmetric) non-linear sigma models. The supersymmetry methodhas a wide range of applications [15]. Examples are the description of singleelectron motion in a disordered or chaotic mesoscopic system [16], chaoticscattering [6], and Anderson localisation [17]. Traditional derivations ofnon-linear sigma models in the supersymmetry formalism rely crucially onhyperbolic Hubbard-Stratonovich transformations. To describe what hy-perbolic Hubbard-Stratonovich transformations are we briefly review thecase of (mathematically trivial) ordinary Hubbard-Stratonovich transfor-mations. These transformations are frequently used throughout condensedmatter field theory. From a mathematical point of view such a Hubbard-Stratonovich transformation consists of applying a Gaussian integral formulabackwards, i.e., introducing additional integrations. Such a scheme convertsa quartic interaction term in the original variables into a quadratic termcoupled linearly to the newly introduced integration variables. The word‘hyperbolic’ indicates a non-compact symmetry group of the original sys-

v

vi INTRODUCTION

tem. In such a situation the standard Gaussian integral formula cannot beapplied due to issues of convergence.1 A solution to this problem was givenby Schafer and Wegner [4]. They found a contour of integration for whichthe Gaussian integral formula holds and convergence is guaranteed. Never-theless the majority of the physics community uses a different contour sug-gested by Pruisken and Schafer [5] which, in contrast to the Schafer-Wegnersolution, preserves the full symmetry of the original system. However, un-til recently there existed no proof of the validity of the Pruisken-Schafertransformation. The main difficulty is that the Pruisken-Schafer domainhas a boundary. This prevents an easy proof similar to the standard Gaus-sian integral and to the Schafer-Wegner domain. Recently, several cases ofthe Pruisken-Schafer transformation have been made rigorous by Fyodorov,Wei and Zirnbauer. Fyodorov [10] gave a proof for pseudounitary symmetryby using methods of semiclassical exactness. After that Fyodorov and Wei[11] proved a variant of the Pruisken-Schafer transformation for the caseof O(1, 1) and O(2, 1) symmetry by direct calculation, and proposed a re-sult for the full O(p, q) case. They conjectured that the Gaussian integraldecomposes into differerent parts that have to be weighted with certain al-ternating sign factors to obtain the right result. This conjecture indicatesthat the Pruisken-Schafer transformation for the pseudoorthogonal case isnot correct in its original form. Finally Fyodorov, Wei and Zirnbauer [12]proved the conjecture by reducing the calculation to the O(1, 1) case andshowing explicitly that all relevant boundary contributions vanish.

The motivation for our work is twofold. First, we want to obtain a bet-ter understanding of the somewhat mysterious alternating sign factors thatappear in the O(p, q) case, and second, we want to generalise the trans-formation to more symmetry classes. The basic idea we follow is that insome sense, the Pruisken-Schafer domain should be a deformation of thestandard Gaussian domain. The problem of the boundary of the Pruisken-Schafer contour is overcome by extending it, such that the integral remainsunchanged and the boundary is moved to infinity. This leads to a proofof a variant of the Pruisken-Schafer transformation for a general symmetrygroup. The proof shows that it is possible to deform the Pruisken-Schaferintegration contour into the standard Gaussian contour without changingthe value of the integral. Actually the same can be done with the Schafer-Wegner contour.

The structure of chapter one is as follows: First we give a more detailedmotivation and a description of the convergence problems one encounterswhen applying the Gaussian integral in case of a non-compact symmetry.Next we discuss a two dimensional example that gives a road map for thegeneral proof. Then we state our result and give its proof. Finally we showhow to obtain the pseudounitary and pseudoorthogonal cases as special cases

1A detailed discussion of this issue is given at the beginning of chapter one.

vii

of the general result.

The second part of this work explores a new method of bosonisationof granular fermionic systems. The terminology ‘granular fermionic’ indi-cates the structure of a fermionic vector model. In the following we listsome examples: The well known Gross-Neveu models [22] and all fermionicmodels having an orbital degeneracy are in this class. An exactly solvabletoy model is the Lipkin-Meshkov-Glick model [21]. A more complicatedexample is the many orbital generalisation of the Hubbard-model. A classof models which is currently intensively studied in mesoscopic physics arearrays of quantum dots or granular metals [24]. Each quantum dot is de-scribed by the universal Hamiltonian, which has a large orbital degeneracy[23]. Note that granularity, or equivalently large orbital degeneracy, impliesthe existence of a natural large N limit. Such large N limits are classicallimits. For Gross-Neveu models this was investigated by Berezin [33] andfor a much larger class of models by Yaffe [34]. In our work we will restrictourselves to discrete (lattice) models that have either orthogonal, unitaryor unitary symplectic symmetry. This contains all relevant possibilities forthe universal Hamiltonian [23]. The term ‘bosonisation’ does not refer tothe well known (non) Abelian bosonisation [19], which is limited to (1 + 1)dimensional models, but rather to the natural geometric approach throughgeneralised coherent state path integrals [35, 36].2 It is interesting to notethat these path integrals lead to a generalised Holstein-Primakoff transfor-mation [18].

The restriction to granular fermionic systems with a classical Lie groupas symmetry group gives access to powerful results from the theory of Howedual pairs [27, 28]. One important tool that relies on the theory of Howedual pairs is the colour-flavour transformation [29, 30]. Within our methodwe put the available structure to use in the calculation of the grand canonicalpartition function of a granular fermionic system. The result we obtain is apath integral representation of the grand canonical partition function of thegranular fermionic system in terms of bosonic, i.e. commuting variables. Therepresentation is essentially a path integral in generalised coherent stateswith certain boundary conditions. However, we cannot apply generalisedcoherent states directly in this context, since this would yield a path integralonly for a subspace of Fock space.

The structure of the second part is as follows: We consecutively discusstwo different derivations of the bosonic path integral representation of thegrand canonical partition function. Furthermore we calculate the contribu-tion of fluctuations in the semiclassical limit in terms of classical quantities.

2There have also been attempts to use coherent state path integrals for loop groups[20] to bosonise (1 + 1) dimensional models.

viii INTRODUCTION

Chapter 1

HyperbolicHubbard-Stratonovichtransformations

1.1 Motivation

Non-compact non-linear sigma models are important and extensively usedtools in the study of disordered electron systems. As mentioned in theintroduction, the corresponding formalism was pioneered by Wegner [3],Schafer & Wegner [4], and Pruisken & Schafer [5]. Shortly afterwards Efetov[2] improved the formalism. He developed the supersymmetry method toderive non-linear sigma models. Many applications of the supersymmetrymethod can be found in the textbook by Efetov [15].

There are different ways to derive non-linear sigma models from micro-scopic models, for an introduction see [8]. The traditional approach is to usea Hubbard-Stratonovich transformation, i.e., a transformation of the form

c0 e−TrA2

=∫

De−TrQ2−2iTrQA |dQ| , (1.1)

where c0 ∈ R. We leave the domain of integration D unspecified for now.|dQ| denotes Lebesgue measure of a normed vector space.

Let us discuss the case of pseudoorthogonal symmetry O(p, q) as an ex-ample. Then A is given by Aij =

∑Na=1 Φa,iΦa,jsjj with s = Diag(1p,−1q)

and Φa,j ∈ R. The Φa,j represent the microscopic degrees of freedom. Us-ing equation (1.1) and integrating out φ gives a description in terms ofthe effective degrees of freedom Q. Thus the task is to find a domain ofintegration D for which identity (1.1) holds and the term exp(−2iTrQA)stays bounded. The latter allows to perform the Φ integrals after applyingidentity (1.1). Note that the real matrices A fulfil the symmetry relationA = sAts. A naive choice of the domain of integration D to keep the term

1

2 CHAPTER 1. HYPERBOLIC HS TRANSFORMATIONS

exp(−2iTrQA) bounded would be the domain of all real matrices satisfy-ing Q = sQts. This choice of D is not valid since then the quadratic formTrQ2 = TrQsQts is of indefinite sign.

Schafer and Wegner [4] found a domain D=SW, and gave a proof thatit solves the problem. Nonetheless another domain D=PS was proposedin later work by Pruisken and Schafer [5]. Until recently the mathematicalstatus of identity (1.1) for D=PS was unclear. The main difficulty in provingidentity (1.1) for D=PS is that the PS domain has a boundary. This preventsan easy proof by completing the square and shifting the contour as is possiblefor the standard Gauss integral and for the SW domain. Nevertheless thePS domain was used in most applications worked out by the mesoscopicphysics community. The reason might be that it inherits the full symmetryof the domain of A matrices.

Recently Fyodorov, Wei and Zirnbauer [10, 11, 12] proved special casesof the Pruisken-Schafer transformation.

In the following we state the result for the O(p, q) case that was obtainedin [12]. Choose D as the subspace of matrices Q = sQts that can be diago-nalised by an element of O(p, q). The domain D can be seen as the union of(p+qp

)subdomains Dσ. The domains Dσ are labeled as follows. Up to a set

of measure zero Q ∈ D has p ‘space-like’ eigenvectors vii≤p with vtisvi > 0and q ‘time-like’ eigenvectors viq<i≤p with vtisvi < 0. Again up to a setof measure zero in D, the eigenvalues of Q can be arranged in decreasingorder. We translate this ordered sequence into a binary sequence by writingthe symbol ‘•’ for space-like and ‘’ for time-like eigenvalues.1 Furthermore,let

|dQ| =∏i≤j

dQij (1.2)

denote flat integration measure on all domains Dσ, and let sgn(σ) be theparity of the number of transpositions • ↔ needed to reduce the binarysequence σ to the extremal form σ0 = • · · · • · · · . Then the followingtheorem [12] holds:

Theorem 1.1. There exists some choice of cutoff function Q 7→ χε(Q) (con-verging pointwise to unity as ε → 0), and a unique choice of sign functionσ → sgn(σ) ∈ ±1 and a constant Cp,q such that

Cp,q limε→0

∑σ

sgn(σ)∫

Dσ

e−TrQ2−2iTrAQχε(Q)|dQ| = e−TrA2(1.3)

holds true for all matrices A = sAts with the positivity property As > 0.

The alternating sign factor was already conjectured in [11]. Note that inthe large N limit only one Dσ contributes to the results. Therefore earlier

1The properties of the domains Dσ are discussed in more detail in [12].

1.2. TWO DIMENSIONAL EXAMPLE 3

works using (1.3) without the alternating sign lead to correct results in thelarge N limit.

In this chapter we show a variant of theorem 1.1 for a general symmetrygroup. The proof shows that it is possible to deform the Pruisken-Schafer(PS) integration contour into the standard Gaussian contour without chang-ing the value of the integral. Actually the same can be done with the Schafer-Wegner (SW) contour. The problem of the boundary of PS is overcome byextending the PS domain in a way that leaves the integral unchanged andmoves the boundary to infinity. The proof also demonstrates the origin ofthe alternating sign factors in equation 1.3

For convenience of the reader, the main ideas of the proof are first il-lustrated in a simple two dimensional example. Then we state the generalresults and give their proof. Finally some applications are discussed. Inparticular, it is discussed how the cases of pseudounitary and orthogonalsymmetry fit into the general setting.

1.2 Two dimensional example

As a first step towards a general theorem of hyperbolic Hubbard-Stratonovichtransformations, we discuss a two dimensional example. In this simplifiedsetting the general result and main ideas of its proof can be nicely illustrated.

First we have to fix the setting. Let e1, e2 be a basis of C2 anddqi(ej) = δij . D denotes the parametrisation of a two dimensional surfacein C2 and dq1 ∧ dq2 is a holomorphic two form on C2. Now consider thesimple Gaussian type integral identity∫

De−q

21+q2

2−2ia1q1+2ia2q2dq1 ∧ dq2 = iπe−a21+a2

2 . (1.4)

At this stage a1 and a2 may be arbitrary complex numbers. In the followingwe discuss different parametrisations of domains of integration D for whichthe identity holds. Eventually this requires imposing additional restrictionson a1 and a2. Since we are integrating differential forms, the domains ofintegration must have an (inner) orientation. Note that we discern betweenitalic D and non italic D. The former denotes the parametrisation of thedomain D.

Euclidean domain of integration

Obviously identity (1.4) holds for the standard Euclidean domain of inte-gration

Euclid : R2 → C2

(r, s) 7→ r e1 + is e2 .


ie2

e1

Figure 1.1: Standard Euclidean domain of integration.

e1

ie1

ie2

Figure 1.2: Schafer-Wegner domain of integration.

Here the orientation comes from choosing an orientation on the domain ofdefinition R2 and declaring Euclid to be orientation preserving. Note thatthe orientation of the two other domains of integration, which we discuss be-low, will be choosen in the same way. The orientation of Euclid is indicatedby a sense of circulation in figure 1.1.

Schafer-Wegner domain of integration

Demanding that ai ∈ R and that a1 > a2 ≥ 0, it can be checked by directcalculation that identity (1.4) also holds for the Schafer-Wegner family ofdomains of integration, which is given by

SW : R2 → C2

(r, s) 7→ r e1 − ib cosh(s) e1 − ib sinh(s) e2 ,

where b > 0.

Pruisken-Schafer domain of integration

The Pruisken-Schafer domain is given by

PS : R2 → R2

(r, s) 7→ r cosh(s) e1 + r sinh(s) e2 . (1.5)


e1

e2

Figure 1.3: Pruisken-Schafer domain of integration. The orientation is in-duced by the parametrisation (1.5).

Identity (1.4) holds for D = PS only in a regularised form. For |a1| > |a2|and χε(~q) = exp(−εq2

2) it can be checked by direct calculation that

limε→0

∫De−q

21+q2

2−2ia1q1+2ia2q2χε(q)dq1 ∧ dq2 = iπe−a21+a2

2 (1.6)

holds.Note that by introducing a Minkowski scalar product B(~a, ~q) = a1q1 −

a2q2 with ~a = (a1, a2)t and ~q = (q1, q2)t, (1.6) can be rewritten to

limε→0

∫De−B(~q,~q)−2iB(~a,~q)χε(~q)dq = iπe−B(~a,~a) . (1.7)

Here we have used dq := dq1 ∧ dq2. B has an O(1, 1) symmetry group,acting on R2. The Pruisken-Schafer domain of integration is invariant underthe action of the group. In this case also the domain of ~a for which (1.6)holds has this invariance. Both domains (for ~a and ~q) are given by forwardand backward lightcones as depicted in figure 1.3, or, put differently, by alltimelike vectors (B(~a,~a) > 0 and B(~q, ~q) > 0).

Moreover, in order for (1.6) to hold it is important that the upper andlower cone (see figure 1.3) have opposite orientations. If one wants to inte-grate Lebesgue measure |dq| rather than a differential form, (1.6) has to bemodified accordingly to a difference of integrals. Defining

g(~q,~a) := e−B(~q,~q)−2iB(~a,~q) ,

identity (1.6) can be reformulated as

limε→0

∫D+

g(~q,~a)χε(q)|dq| − limε→0

∫D−

g(~q,~a)χε(q)|dq| = iπe−a21+a2

2 ,

where D+ stands for the upper and D− for the lower cone in figure 1.3.It was already mentioned in the introduction that for more general cases

it is very difficult to prove identity (1.6) for the Pruisken-Schafer domain bydirect calculation. The idea of the proof for the general case is now discussedin the two dimensional case.


Main idea of the proof

The main idea is to deform the PS domain into the Euclidean domain with-out changing the value of the integral. As an easy example for such adeformation scheme we can use the SW domain of integration. Consider thedeformation (or homotopy) given by

DSW : [0, 1]×R2 → C2

(t, r, s) 7→ re1 + ib(1− t) cosh(s)e1 + ib sinh(s)e2 ,

which is just a smooth projection onto the plane spanned by e1 and ie2.Note that DSW (t = 1) = Euclid and that the parametrisation given byDSW is a nice integration chain. Hence, ∂DSW = Euclid − SW and wecan apply Stokes theorem:

0 =∫DSW

d (g(~q,~a)dq1 ∧ dq2)︸︷︷︸=0

=∫∂DSW

g(~q,~a)dq1 ∧ dq2 .

Thus we have∫SW

g(~q,~a)dq1 ∧ dq2 =∫Euclid

g(~q,~a)dq1 ∧ dq2 = iπe−B(~a,~a) . (1.8)

The PS domain has a boundary, which seems to prevent an analogous proofof (1.6).

Proof

The following proof of (1.6) for a1 > a2 ≥ 0 mimics the proof of the higherdimensional relatives of (1.6). Therefore the proof should be seen as a roadmap for the more complicated proof in the next section.

A suitable parametrisation: A different parametrisation of the PSdomain is obviously given by

PS : [−1, 1]× R→ R2

(h, x) 7→ x(e1 + he2) .

The boundary operator ∂ gives a nonzero result on [−1, 1]. The other con-tributions vanish since the integrals to be considered are exponentially con-vergent for ε > 0. We therfore have ∂PS = PS(1)− PS(−1).

Extending the PS domain: Before deforming the PS domain, theboundary problem has to be dealt with first. The idea is to attach half-planes to the boundary lines spanned by e1 ± e2 as illustrated in figure 1.4.Here it is again crucial that the upper and lower cone have opposite ori-entations to allow the attachment of the halfplanes in a consistent way. Aparametrisation of the halfplanes is given by


e1 + e2e1 + e2

i(e1 + e2)

attach

Figure 1.4: Attaching halfplanes that do not contribute to the integral.

hp± : R+ ×R→ C2

(h, x) 7→ ±xe± − ihe± ,

where e± = e1 ± e2. A good motivation for this special choice is that

dq1 ∧ dq2(e±, ie±) = idq1 ∧ dq2(e±, e±) = 0

and hence the halfplanes do not contribute to the integral. This is notyet a completely rigorous argument since existence and convergence of theintegral still needs to be discussed. The extension of PS is defined as ePS :=PS + hp+ + hp−, which is to be understood as a sum of integration chains.

Equivalence of PS and Euclid: The deformation is given by

DPS : [0, 1]× [−1, 1]×R→ R2

(t, h, x) 7→ x(e1 + (1− t)he2) ,

Dhp± : [0, 1]×R+ ×R→ C2

(t, h, x) 7→ ±x[e1 ± (1− t)e2]− ih[(1− t)e1 ± e2] ,

which defines then DePS = DPS+Dhp+ +Dhp−. For t = 1 the deformedPS surface degenerates into a line and both halfplanes hp± are projectedinto the plane spanned by e1 and ie2 as shown in figure 1.5. Note thatDhp±(t = 1) : (h, x) 7→ ±xe1∓ ihe2. Thus we have DePS(t = 1) = Euclid.

Now, we want to apply Stokes’ theorem. We use ∂DePSε = Euclid −DePS(ε), where DePSε := DePS|t∈[ε,1]. The idea is to deform Euclid, asfar as convergence of the integral allows, into PS:∫

DePS(ε)g(~q,~a)dq = −

∫DePSε

g(~q,~a)d(dq) +∫Euclid

g(~q,~a)dq

=∫Euclid

g(~q,~a)dq . (1.9)


e1

ie2

Figure 1.5: Deformation of PS to Euclid. The two halfplanes are the (de-formed) attached surfaces and the cones are deformed into the vertical line.

Next we have to discuss the limit ε→ 0 carefully.First consider the PS part. Here, we may apply Fubini’s theorem and

perform the x integration first, since for ε > 0 the integral is exponentiallyconvergent. We define

IPS(ε, h) :=√

π

1− h2(1− ε)2e− (a1−a2h(1−ε))2

1−h2(1−ε)2 .

Then we obtain

limε→0

∫DPS(ε)

g(~q,~a)dq = limε→0

∫ 1

−1dh IPS(ε, h) .

The limit limε→0 IPS(ε, h) is uniform, since (a1 − a2h(1 − ε))2 > 0 for 1 ≥ε ≥ 0 and a1 > a2 ≥ 0. Thus the h integral and limε→0 commute.

In particular, this is also true if we replace IPS(ε, h) by

IPS(ε, h) :=√

π

1− h2(1− ε)2e− (a1−a2h)2

1−h2(1−ε)2 .

Then we have the following series of equalities:

limε→0

∫DPS(ε)

g(~q,~a)dq =∫ 1

−1dh lim

ε→0IPS(ε, h)

= limε→0

∫ 1

−1dh IPS(ε, h)

= limε′→0

∫PS

g(~q,~a)χε′(q)dq ,

where we shift the ε dependence of the domain of integration to the inte-grand by introducing a regulating function χε′(q) = exp(−ε′q2

2). To be moreprecise, we identify ε′ = 2ε− ε2.

It remains to show that the contribution from the hp± parts vanish. Thisis done using similar arguments as above. Consider

limε→0

∫Dhp±(ε)


∫ ∞0

dh Ihp±(ε, h) ,

1.3. GENERAL SETTING AND THEOREM 9

where we define

Ihp±(ε, h) :=√

π

1− (1− ε)2e− (±a1−(1−ε)a2)2

1−(1−ε)2 e−2h((1−ε)a1∓a2)e−h2(2ε−ε2) .

The last two factors ensure exponential convergence in h. Hence the integralexists and

limε→0

Ihp±(ε, h) = 0

holds uniformly in h. Thus we can conclude that

limε→0

∫Dhp±(ε)

g(~q,~a)dq = 0 .

The proof is finished, since we now have

limε→0

∫DePS(ε)


∫PS

g(~q,~a)χε(q)dq . (1.10)

1.3 General setting and theorem

In this section we present our results in a rather general form. First, wedescribe the setting and then we state our theorem. Let us note in advancethat appendix A contains a systematic discussion of the structures that areused to formulate the theorem and its proof.

All constructions take place in gl(n,C), the space of all complex n ×n matrices. The following results also apply to the case where gl(n,C)is replaced by a complex Lie subalgebra of gl(n,C). Let s ∈ gl(n,C) behermitian with the property s2 = 1. s leads to two involutions2 θ(X) =sXs−1 and γ(X) = −sX†s−1 on gl(n,C). In addition we assume that weare given some involutions τi on gl(n,C), which commute with each otherand with θ and γ. Then we can define a subspace Q of gl(n,C) as

Q = Q ∈ gl(n,C)|Q = −γ(Q) and ∀i : Q = σiτi(Q) , (1.11)

where σi ∈ ±1 and the τi have to be such that s ∈ Q. The Lie algebra ofthe relevant symmetry group of Q is given by3

g = X ∈ gl(n,C)|X = γ(X) and ∀i : X = τi(X) . (1.12)

The decomposition of Q into the plus and minus one eigenspaces of θ givesa decomposition into the hermitian and antihermitian parts denoted by Q+

and Q−. Similarly θ gives the decomposition g = k⊕ p, where k is the plus2In this context involution means an involutive automorphism of Lie algebra.3See also section A.1.2 in appendix A.


one eigenspace and p the minus one eigenspace. The commutation relationsof these spaces are4

[k, k] ⊂ k , [k, p] ⊂ p , [p, p] ⊂ k , [Q+,Q−] ⊂ p ,[Q±,Q±] ⊂ k , [k,Q±] ⊂ Q± , [p,Q±] ⊂ Q∓ .

(1.13)

Then the parametrisation of the Pruisken-Schafer domain is given by

PS : p⊕Q+ → Q(Y,X) 7→ eYXe−Y . (1.14)

The parametrisation of the Euclidean domain is given by

Euclid : Q− ⊕Q+ → QC

(Y,X) 7→ X + iY ,

where QC = Q⊕ iQ. The parametrisation of the Schafer-Wegner domain isgiven by

SW : p⊕Q+ → QC (1.15)

(Y,X) 7→ X − ibeY se−Y , (1.16)

where b is a positive real number. The orientation for PS, Euclid and SWis provided by choosing an orientation of the domain of definition. Thisinduces an orientation on the corresponding domain of integration.

Theorem 1.2. If in the setting above g is the direct sum5 of a semisimpleand an Abelian Lie algebra and A ∈ Q with As > 0, then

limε→0

∫PS

e−Tr(Q2)−2iTr(QA)χε(Q)dQ = c e−Tr(A2) (1.17)

holds. Here, χε(Q) = exp( ε4 Tr(Q − θQ)2) is a regulating function and dQdenotes a constant volume form on Q. c ∈ C \ 0 is a constant that doesnot depend on A.

The result will be proved by showing that the PS domain can first beextended and then deformed into a standard Euclidean integration domainwithout changing the value of the integral. Therefore we view DQ as holo-morphic dimQ form onQC. In addition it will be shown that the SW domaincan also be deformed into this Euclidean integral. Hence the PS and SWdomains are contour deformations of the same simple Euclidean Gaussiandomain.

The following corollary is the analogue of corollary 1 in [12]:4See section A.1.3 in appendix A.5This means in particular that the semisimple and the Abelian part commute with

each other.

1.4. PROOF OF THE THEOREM 11

Corollary 1.1. Let k⊕Q+ be the direct sum of a semisimple and an AbelianLie algebra and let h be a maximal Abelian subalgebra of Q+. Furthermorelet g be semisimple and define G = exp(p) exp(k)6. Then

limε→0

∫h

(∫Ge−2iTr(gλg−1A)χε(gλg−1)|dg|

)e−Trλ2

J ′(λ)|dλ| = c e−Tr(A2) ,

holds, where

J ′(λ) =∏

α∈Σ+(p⊕Q−,h)

α(λ)dα∏

α∈Σ+(k⊕Q+,h)

|α(λ)dα | .

Σ+(V, h) denotes the sets of positive weights with respect to the adjoint actionof h with weight spaces in V .7 dα are the dimensions of the weight spaces.|dg| denotes Haar measure on G and |dλ| denotes Lebesgue measure on thevector space h. c ∈ C \ 0 is a constant that does not depend on A.

The following corollary is the analogue of theorem 1 in [12]:

Corollary 1.2. If the parametrisation PS is nearly everywhere injectiveand regular, then

limε→0

∫ImPS

e−Tr(Q2)−2iTr(QA) χε(Q) sgn(J ′(λ)) |dQ| = c′e−Tr(A2)

holds. ImPS denotes the image of PS and the mapping from ImPS to h

sending Q to λ is well defined up to a set of measure zero. |dQ| denotesLebesgue measure on Q. c′ ∈ C \ 0 is a constant that does not depend onA.

1.4 Proof of the theorem

For simplicity we restrict ourselves to the case where g is semisimple. Theextension to the more general case is straightforward. The proof is dividedinto three parts. The first part, 1.4.1, contains the derivation of a newparametrisation of the PS domain, which makes its boundary visible in thedomain of definition. Decomposing this parametrisation suitably as a sum ofintegration cells one obtains an adequate description of the boundary of thePS domain. The second part of the proof, 1.4.2, deals with the extension ofthe PS domain to a domain ePS without boundary. First we identify gooddirections into which the PS domain can be extended. Then an extensionof PS that does not change the value of the integral is given. Finally insection 1.4.3 we construct a deformation DePS of the extended PS domainto the Euclidean domain. The deformation satisfies ∂DePS = Euclid−ePS.

6G is the unique analytic subgroup of GL(n, C) with Lie algebra g.7See also section A.1.4 in appendix A.


Essentially we want to make rigorous the following schematic application ofStokes: ∫

PSg(Q,A)dQ =

∫ePS

g(Q,A)dQ

= −∫DePS

d(g(Q,A)dQ)︸︷︷︸=0

+∫Euclid

g(Q,A)dQ ,

where we define g(Q,A) := e−Tr(Q2)−2iTr(QA). Note that the first term inthe second line is identically zero since g(Q,A) is holomorphic in Q. At thispoint a warning is in order: In this form the upper expressions do not makesense. In order for the integrals over PS, ePS and DePS to exist we haveto include some regularisation. This delicate issue is discussed in detail inthe last part of section 1.4.3.

1.4.1 A suitable parametrisation of the PS domain

First, we perform a series of reparametrisations to derive a more convenientparametrisation which allows to use the geometric intuition from the twodimensional example for the PS domain and its boundary. Finally we de-compose the parametrisation into different parts, allowing the application ofStokes’ theorem. In the following we use standard results from Lie theory.A good reference is [14]. In addition appendix A gives a detailed descriptionof the constructions we use.

Reparametrisation I: Decomposition of p

The goal of the next three reparametrisations is to evaluate PS(Y,X) =Ad(eY )X in more detail. Key to this is choosing a maximal Abelian sub-algebra a in p, whose adjoint action on Q = Q+ ⊕Q− can be diagonalisedsimultaneously.8 To begin with, we decompose the parameter space p to seethe algebra a. Therefore we define the compact group K := exp(k) and thecentraliser ZK(a) of a in K. Furthermore ao+ ⊂ a denotes the interior of afixed Weyl chamber.9 Consider the mapping

RI : ao+ ×K/ZK(a)→ p

(H, [k]) 7→ kHk−1 .

RI is obviously well defined. In appendix A.2.1 it is shown that RI isinjective and regular. Hence RI is a diffeomorphism onto. Note, that p \Im(RI) is a set of measure zero since p = ∪k∈Kkak−1 10 and Im(RI) =∪k∈Kk(a\∪α kerα)k−1 where α are the restricted roots with respect to a.11

8See section A.1.4 in appendix A.9See [14] for a definition and properties of Weyl chambers.

10This standard result, which can be found in [14], is used without further explanation.11See section A.1.4 in appendix A.


To be precise we want to use the parametrisation

PS RI : a+ ×K/ZK(a)×Q+ → Q

(H, [k], X) 7→ ekHk−1Xe−kHk

−1.

The orientation of PS is given by an orientation of p ⊕ Q+. Declaring RIto be orientation preserving induces an orientation on ao+×K/ZK(a)×Q+.To keep the notation simple we call each new parametrisation again PS.

Reparametrisation II: Twisting K/ZK(a) and Q+

In this section we prepare further evaluation of the a action in the nextsubsection. Consider the reparametrisation

RII : K ×ZK(a) Q+ → K/ZK(a)×Q+

[kz−1, zXz−1] 7→ ([k], kXk−1) .

Note that z ∈ ZK(a) in the expression [kz−1, zXz−1] indicates group actionsof ZK(a) on K and onQ+. These group actions are used to define the bundleK ×ZK(a) Q+. The inverse of RII is given by

R−1II : K/ZK(a)×Q+ → K ×ZK(a) Q+

([k], X) 7→ [k, k−1Xk] .

RII is obviously a diffeomorphism and therefore can be used as a reparametri-sation to obtain

PS RII : ao+ ×K ×ZK(a) Q+ → Q

(H, [kz, z−1Xz]) 7→ ekHk−1kXk−1e−kHk

−1

= keHXe−Hk−1

as new parametrisation.

Reparametrisation III: Decomposition of Q+

The weight decomposition of Q with respect to the adjoint action of a isgiven by

Q = Q0 ⊕⊕

α∈Σ+(Q,a)

(Qα ⊕Q−α) .

Here Σ+(Q, a) denotes the set of posive weights. Defining

Q+,α := Fixθ(Qα ⊕Q−α) and Q+,0 = Fixθ(Q0) ,


we obtain the decomposition

Q+ = Q+,0 ⊕⊕

α∈Σ+(Q,a)

Q+,α .

For more details and properties of this decomposition see appendix A.1.4.Since ZK(a) is a subset ofK, and, by definition, commutes with the ad(a)

action on Q, the decomposition is compatible with the bundle structure ofK ×ZK(a) Q+. Again, the reparametrisation

RIII : a+ ×K ×ZK(a)

(Q+,0 ⊕

⊕α∈Σ+(Q,a)

Q+,α

)→ a+ ×K ×ZK(a) Q+

(H, [kz, z−1Mz, z−1Xαz]) 7→(H,[kz, z−1

(M +

∑α∈Σ+(Q,a)

Xα

)z])

,

is an orientation preserving diffeomorphism.We define a mapping φ : Q+ → Q− implicitly through

[H,Xα] = α(H)φ(Xα) and [H,φ(Xα)] = α(H)Xα (1.18)

for all H ∈ a. See also appendix A.1.3. Using (1.18) a short calculationgives

ead(H)Xα = cosh(α(H))Xα + sinh(α(H))φ(Xα) .

Moreover, the parametrisation can be rewritten to

PS RIII : a+ ×K ×ZK(a)

(Q+,0 ⊕

⊕α∈Σ+(Q,a)

Q+,α

)→ Q

(H, [k,M,Xα])

7→ Ad(k)[M +

∑α∈Σ+(Q,a)

[cosh(α(H))Xα + sinh(α(H))φ(Xα)

]].

In the following PS RIII is denoted simply by PS.

Reparametrisation IV: Transfer boundary to a+

As a motivation for the next reparametrisation of Q+,α one might imag-ine the coordinate line belonging to λH ∈ a+ as a hyperbola. We wantto have a simple Euclidean picture of the situation. Thus we change ourparametrisation in a way that these coordinate lines are straight lines, seefigure 1.6. Most importantly such a reparametrisation simplifies the viewon the boundary of the PS domain, as is discussed in the next subsection.


Q+,α

Q−,α

Q+,α

Q−,α

Figure 1.6: Motivation for the third reparametrisation step. The dashedlines are coordinate lines of λH ∈ a+.

Thus, the fourth reparametrisation we use is given by

RIV : a+ ×K×ZK(a)

(Q+,0 ⊕

⊕α∈Σ+(Q,a)

Q+,α

)→ a+ ×K ×ZK(a)

(Q+,0 ⊕

∑α∈Σ+(Q,a)

Q+,α

)(H, [k,M,Xα]) 7→

(H,[k,M,

1cosh(α(H))

Xα

]),

which is also an orientation preserving diffeomorphism. We obtain

PS RIV : a+ ×K ×ZK(a)

(Q+,0 ⊕

⊕α∈Σ+(Q,a)

Q+,α

)→ Q

(H,[k,M,Xα]) 7→ Ad(k)[M +

∑α∈Σ+(Q,a)

[Xα + tanh(α(H))φ(Xα)

]],

(1.19)

which we again call PS in the following.

Boundary of the PS domain

In this subsection we explain why parametrisation (1.19) is useful to get anintuition for the geometry and especially the boundary of the PS domain.This discussion is not meant to be rigorous but motivates the next steps ofthe proof.

The geometry can be described using Tr(XY †) as an Ad(K)-invariantscalar product on Q. Thus for the moment we forget about the Ad(K)


Q+,α

Q−α

Figure 1.7: Interior of the PS domain. The rest is generated by the Kaction.

action and restrict ourselves to the inner part

iPS : a+ ×(Q+,0 ⊕

⊕α∈Σ+(Q,a)

Q+,α

)→ Q

(H,M,Xα) 7→M +∑

α∈Σ+(Q,a)

[Xα + tanh(α(H))φ(Xα)

].

Note that the different Q+,α and also Q−,α are all orthogonal to each other.Hence we consider only one α ∈ Σ+(Q, a) at a time:

Xα + tanh(α(H))φ(Xα) ,

for which figure 1.7 is a good two dimensional picture. Except for theadditional presence of the Weyl group, figure 1.7 is in agreement with figure1.3 of the two dimensional example.

It is clear that the boundary is reached when some α(H) goes to ±∞ andhence tanh goes to ±1. To put it differently, the boundary of the domain ofintegration can be reached through a limit in the parameter space a+. Notethat acting with K on the boundary, identified in the inner part of the PSparametrisation, the full boundary is generated.

Since we want to get rid of the boundary by attaching halfplanes to thePS domain, we want to give a parametrisation which reaches all boundarypoints. This implies performing the limit explicitly and thus making theboundary visible in the domain of definition.

Decomposition of the parametrisation PS

The problem we face is to perform the limit in a+ in a well defined way.Therefore we have to discuss the weights α ∈ Σ+(Q, a) in more detail.

First note that α(H) might change sign on a+ since a+ is defined withrespect to the restricted roots β ∈ Σ(g, a). Within this subsection α willalways denote a weight in Σ+(Q, a).


The main idea is to decompose the PS parametrisation by decomposinga+ into different cones on which sgn(α(H)) stays constant or α(H) goes tozero for all α ∈ Σ+(Q, a).

The closures of the connected components of a+ \ ∪α ker(α) are pointedpolyhedral cones, whose edges lie in the intersections of hyperplanes definedby the kernels of the weights in Σ+(Q, a). Let us consider one of thesepointed cones. It can also be defined as an intersection of halfspaces or asnon-negative linear combination of some generators Hi ∈ a. The generatorsare the edges of the cone. In general the number of generators might begreater than n := dim a. But each pointed cone can be triangulated (withoutintroducing new vertices) into simplicial cones, i.e., cones where the numberof generators equals dim a. See for example [13]. Now we fix triangulationsfor each original pointed polyhedral cone. Thus we obtain a decompositionof a+ into simplicial cones, which we denote by a+,c and

a+ =⋃c∈C

a+,c , (1.20)

where C is an index set for the different cones. Denote the generators/edgesof a+,c by Hi,c. Then we can represent H ∈ a+,c uniquely as

H =n∑i=1

hiHi,c (1.21)

with coefficients hi ∈ R+. The important thing is that the sign of all α ona given simplicial cone stays constant. However it is still allowed that αvanishes at the boundary of the simplicial cone. Thus we decompose ourparametrisation as follows:

PS =∑c∈C

PS|a+,c×K×ZK (a)(Q+,0⊕Lα∈Σ+(Q,a)Q+,α) .

In the following we hide the index c of Hi,c and use the notation H =∑i h

iHi for H ∈ a+ which implies the choice of Hi as described above. Byconstruction we then have hi ≥ 0.

Reparametrisation V: Making the boundary visible

To make the boundary visible in the domain of definition we define

aB,c =

n≡dim a∑i=1

hiHi ∈ a+,c|∀i : 0 ≤ hi ≤ 1

.

and use the reparametrisation

RV,c : aoB,c → a+,c

n∑i=1

hiHi 7→n∑i=1

hi

1− hiHi ,


which is an (orientation preserving) diffeomorphism onto for each simplicialcone. The reparametrisation is visualised in figure 1.8. Note that there isan obvious diffeomorphism between [0, 1]n and aB,c. Defining for each conec ∈ C

PSc : [0, 1]n ×K ×ZK(a)

(Q+,0 ⊕

⊕α∈Σ+(Q,a)

Q+,α

)→ Q

(hi,[k,M,Xα]) 7→

Ad(k)[M +

∑α∈Σ+(Q,a)

(Xα + tanh

(∑i

hi

1− hiα(Hi)

)φ(Xα)

)], (1.22)

we have PS =∑

c∈C PSc as an equation of integration chains. It is impor-tant to keep in mind that [0, 1]n is essentially aB,c, which can be seen as atruncated cone in a+.

In the following we want to give the notion ‘boundary of the PS domain’a precise meaning. For integration cells, i.e., differentiable mappings definedon a cube, the boundary operator ∂ is defined as usual. ∂ can also be appliedto integration chains, i.e. formal linear combinations of cells. In principlewe would have to decompose each PSc into cells to apply ∂. In the followingwe argue that we can treat each PSc effectively as cell with the boundaryoperator ∂ acting just on the [0, 1]n part of the domain of definition.

First we have to show that PSc can be extended to a neighbourhoodof [0, 1] on which it is still differentiable. This property is included in thedefinition of an integration cell. It is needed to define the orientation of theboundary. Let us first concentrate on the tanh term in the PSc parametri-sation. Since

limhj→1

∂hj tanh[ n∑i=1

hi

1− hiα(Hi)

]= 0 , (1.23)

generalises to all higher (and mixed) partial derivatives, we extend PSc forhi > 1 by setting it constant in that direction. To be more precise we definefor hi > 0

PSc(. . . , hi, . . . ) := PSc(. . . , 1, . . . ) .

For hi < 0 just analytically continue the parametrisation PSc. Using (1.23)it can be easily shown that the extension of PSc to a neighbourhood of[0, 1]n is differentiable.

Now we present an argument that we can treat each PSc effectively ascell and that it is enough to let the boundary operator ∂ act on the [0, 1]n

part of the domain of definition. Since K is a closed compact manifold itsuffices to discuss boundary contributions arising from a decomposition ofQ+,0 ⊕

∑α∈Σ+(Q,a)Q+,α into cells. Inspecting our parametrisation we see


RV

Figure 1.8: su(2, 2) example for RV mapping aB on the left hand side toa+ on the right side. In this example there is only one simplicial cone, i.e.|C| = 1.

that going to infinity in the domain of definition implies going to infinityin the domain of integration. In section 1.4.3 we show that the integrandconverges exponentially on this domain and hence all possible boundarycontributions vanish.

Only the part [0, 1]n, or equivalently, the part aB,c of the domain ofdefinition, leads to a nontrivial boundary. The boundary parts coincidingwith boundaries of a+ are of codimension at least two, and thus do notcontribute. For a detailed argument concerning this point see appendixA.2.3.

1.4.2 Extending the PS domain

In this section we construct an extension of the PS domain that has norelevant boundary. This means we have to attach additional domains tothe boundary of the PS domain. The idea is to attach a halfline to eachboundary point. The direction of this halfline should be a convergent one,and it should also guarantee that the attached domain does not contributeto the integral when integrated against f(Q) dQ. First we determine such adirection, and then give a parametrisation of the attached domains. In thefollowing it is often convenient to view B(X,Y ) := Tr(XY ) as a bilinearform on QC.

Good directions

We want to extend the PS domain from the boundary into an imaginary nulldirections Ei. The terminology ‘null direction’ conveys two things. First Eiis a null direction in the sense that B(Ei, Ei) = 0. And second the extensionin this direction does not contribute to the integral since the volume formvanishes for these directions. Ei shall also be a convergent direction. It isreasonable to expect that the term exp(−2iB(Q,A)) guarantees convergencein this situation if

< [iB(Ad(k)Ei, A)] = <[iTr(s−1 Ad(k)EiAs)] > 0 . (1.24)


In order for (1.24) to hold it suffices that isAd(k)Ei ≥ 0 and Ei 6= 0, sinceAs > 0 by assumption. For our definition of Ei below it is important thatfor Y ∈ p

s−1eY se−Y = e−2Y > 0

holds. The natural choice for Ei is

Ei := −2i limt→∞

Ad(etHi)smaxα∈Σ+(Q,a) e|α(tHi)|

. (1.25)

Note that ZK(a) acts trivially on Ei. Let us also mention again that we havehidden the dependence of Hi on c ∈ C. Thus Ei also depends on c ∈ C.

It is instructive to give a more explicit form of Ei. Since s ∈ Q+ we havethe decomposition

s = Ms +∑

α∈Σ+(Q,a)

Xs,α , (1.26)

where Ms ∈ Q+,0 and Xs,α ∈ Q+,α. A short calculation gives

Ad(eH)s = Ms +∑

α∈Σ+(Q,a)

cosh(α(H))Xs,α + sinh(α(H))φ(Xs,α) .

This shows that the limit in (1.25) exists. In addition Ei 6= 0 because a →Q−, H 7→ [H, s] is injective. The properties of this mapping are discussed indetail in appendix A.1.3. Note also that Ei depends on the chosen simplicialcone c, and that B(Ei, Ei) = 0. Most importantly inequality (1.24) holds(even without taking the real part).

is−1Ei can be regarded as an orthogonal projection on the weight space(in the vector space on to which g acts) with the largest eigenvalue withrespect to the eHi action.

Parametrisation of the extension

In this section we suggest an extension of the PS domain and then checkthat it has all the desired properties. The extended PS domain (ePS) isparametrised by

ePS : a+×K ×ZK(a)

(Q+,0 ⊕

∑α∈Σ+(Q,a)

Q+,α

)→ QC

(H, [k,M,Xα]) 7→

Ad(k)[M +

∑α∈Σ+(Q,a)

[Xα + tanh

(∑j

ξ(hj)α(Hj))φ(Xα)

]

+n∑j=1

Θ(hj − 1)(hj − 1)Ej],


where we use

ξ(h) :=

h1−h h < 1∞ h ≥ 1

Θ is the step function. The attached surface starts as soon as some hi > 1.The limit contained in the expression tanh(

∑j ξ(h

j)α(Hj)) is well de-fined because sgn(α(Hj)) is the same for all i with α(Hi) 6= 0. Note thatthe mapping ePS is well defined since ZK(a) acts trivially on the Ei.

In the following we decompose the parametrisation ePS into differentpieces, which can be treated as integration cells. Remember that we al-ready have the corresponding decomposition PS =

∑c∈C PSc. For ePS the

situation is slightly more involved as we have to account explicitly for thelimit being taken, i.e., which hi are larger than one. The different possibil-ities are characterised by subsets L ⊂ 1, 2, . . . , n = dim a. The extendedparametrisation decomposes into several pieces ePSL,c. The situation isvisualised in figure 1.9. To be able to write down ePSL,c explicitly we define

ΣL,6= := α ∈ Σ+(Q, a)|∃i ∈ L : α(Hi) 6= 0

and

ΣL,= := α ∈ Σ+(Q, a)|∀i ∈ L : α(Hi) = 0 .

Then we have for each L ⊂ 1, 2, . . . , n = dima and c ∈ C

ePSL,c : [0, 1]n−|L| × [1,∞)|L| ×K ×ZK(a)

(Q+,0 ⊕

∑α∈Σ+(Q,a)

Q+,α

)→ QC

(hi, [k,M,Xα]) 7→

Ad(k)[M +

∑α∈ΣL, 6=

(Xα + sgn(α(Hi))φ(Xα))

+∑

α∈ΣL,=

[Xα + tanh

( n∑j=1


]+∑j∈L

(hj − 1)Ej].

Next we present an argument that ePSL,c can be treated as an integrationcell and that the boundary operator ∂ only acts on [0, 1]n−|L| × [1,∞)|L|.First we discuss the extension of ePSL,c to a neighbourhood of [0, 1]n−|L| ×[1,∞)|L|. For i /∈ L and hi > 0 we define

ePSL,c(. . . , hi, . . . ) := ePSL,c(. . . , 1, . . . ) ,

and for hi < 0 or i ∈ L and hi < 1 we analytically continue the parametrisa-tion ePSL,c. The boundary operator ∂ applied to ePSL,c is evaluated using


H1

H2

L = ∅

L = 2

L = 1

L = 1, 2

Figure 1.9: This figure shows a+ for g = su(2, 2). a+ is decomposed intothe domains of definition for the different mappings ePSL,c in a+ with L ⊂1, 2.

exactly the same reasoning as for the PSc mappings. Thus ∂ acts only on[0, 1]n−|L| × [1,∞)|L|. To see that

∂ePS =∑c∈C

∑L⊂1,2,...,n

∂ePSL,c = 0

holds, note that the different integration cells ePSL,c fit together by def-inition, i.e., the induced orientation on the boundaries between two neigh-bouring cells is just opposite. In section A.2.3 in appendix A it is shownthat the contributions from ∂a+ are of codimension at least two. Thus acontribution can only come from hi going to infinity, but the integral isconvergent and hence these terms do not contribute either.

To get some intuition for the situation it is useful to note that thehalflines which are glued to boundary points of the PS domain point intoa direction within ⊕α∈Σ(Q,a)iQα, and hence cannot coincide with tangentvectors to PS which live in Q.

Extensions are nullsurfaces

Now we want to motivate that the extension does not contribute to theintegral. The following argument is not rigorous as it does not refer toexistence and convergence of the integrals involved.

The holomorphic volume form gives zero if two linearly dependent (overC) vectors are inserted. Thus it is enough to show that we can find twotangent vectors of the extension which are linearly dependent over C. Lethj > 1, then one tangent vector is Ad(k)Ej(H). The latter can be expanded


as follows:

Ej = −i∑

α∈Σj,6=

eαj (Xs,α + sgn(α(Hj))φ(Xs,α)) , (1.27)

where eαj ∈ 0, 1. Within the boundary parametrisation the following termsare contained:

Ad(k)∑

α∈Σj,6=

(Xα + sgn(α(Hj))φ(Xα)) .

It is clear that by differentiation in∑

α∈Σ+(Q+,a)Q+,α in the direction of∑α∈Σi, 6=

eαi Xs,α we obtain a tangent vector parallel (over C) to Ad(k)Ei.This implies that the volume form vanishes, and that the extension of PSdoes not contribute to the integral. This argument does not incorporatethe function which is integrated, but only on the integration chain and thevolume form.

1.4.3 Equivalence of PS and Euclid

Finally we want to show that the integral over PS equals the integral overEuclid. First we give a deformation of ePS into Euclid. Then we applyStokes’ theorem. To get the desired equation we first show the existence ofthe appearing integrals with regularisation ε > 0. A careful discussion ofthe limit ε going to zero yields the theorem.

Deformation of ePS into Euclid

The idea is to deform ePS into the subspace Q+ ⊕ i[p, s] of QC where B ispositive definite. Note that in appendix A.1.3 it is shown that [p, s] = Q−.Along the deformation we have to show that the integral remains convergent,so that no boundary terms at infinity are generated.

The deformation is given by

DePS : [0, 1]× a+ ×K ×ZK(a)

(Q+,0 ⊕

∑α∈Σ+(Q,a)

Q+,α

)→ QC ,

(t,H,[k,M,Xα]) 7→

Ad(k)[M +

∑α∈Σ+(Q,a)

(Xα + (1− t) tanh

( n∑j=1


)]︸︷︷︸

Ξ

+ Ad(k)[ n∑j=1

Θ(hj − 1)(hj − 1)(Ej − tθ(Ej) + t∆Xs,j)]

︸︷︷︸Υ

, (1.28)


where we use

∆Xs,i := −2∑

α∈Σ+(Q,a)

α(Hi)iφ(Xs,α)− (Ei − θ(Ei)) .

Note also that DePS is well defined, since the involution θ commutes withthe action of the centraliser ZK(a). To proceed we decompose DePS simi-larly as ePS:

DePSL,c : [0, 1]1+n−|L| × [1,∞)|L| ×K ×ZK(a)

(Q+,0 ⊕

∑α∈Σ+(Q,a)

Q+,α

)→ QC

(t,hi, [k,M,Xα]) 7→ Ad(k)[M

+∑

α∈ΣL, 6=

(Xα + (1− t) sgn(α(Hi))φ(Xα))

+∑

α∈ΣL,=

[Xα + (1− t) tanh

(∑j


]+∑j∈L

(hj − 1)(Ej − tθ(Ej) + t∆Xs,j)].

Using similar reasoning as for ePSL,c each parametrisation DePSL,c can beseen as integration cell with ∂ acting only on the [0, 1]1+n−|L| × [1,∞)|L|

part. In particular we have

∂DePS =∑c∈C

∑L⊂1,...,n

∂DePSL,c = DePS(t = 1)−DePS(t = 0) ,

where we used in the last equality that the contributions from boundariesof a+ vanish.12

Inspecting our previous arguments it is clear that for t ∈ [0, 1), DePSL,c(t)are integration chains with the same properties as ePSL,c. For t = 1 andL 6= 1, . . . , n the parametrisation DePS(1) degenerates, i.e., for i /∈ L wehave ∂hiDePS(1) = 0. Hence we have the following equation for integrationchains:

DePS(1) =∑c∈C

DePS1,...,n,c(1) .

In the following we establish the connection between DePS(1) and Euclid.Therefore it is useful to introduce the mapping

ψ : Rn+ → [1,∞)n

(. . . , hi, . . . ) 7→ (. . . , hi + 1, . . . ) .

12See section A.2.3 in appendix A.


The precise statement we show is:

Euclid =∑c∈C

DePS1,...,n,c(1) (ψ, id) R−1III R

−1II R

−1I ,

where id denotes the identity on K×ZK(a) (Q+,0⊕∑

α∈Σ+(Q,a)Q+,α). Since[p, s] = Q−, the following calculation suffices:

DePS(1, ψ(H), [k,M,Xα]) = Ad(k)(M +

∑α∈Σ+(Q,a)

Xα − i∑

α∈Σ+(Q,a),i

2hiα(Hi)φ(Xs,α))

= Ad(k)(M +

∑α∈Σ+(Q,a)

Xα − 2i∑

α∈Σ+(Q,a)

[H,Xs,α])

= Ad(k)(M +

∑α∈Σ+(Q,a)

Xα

)− 2iAd(k)[H, s]

= Ad(k)(M +

∑α∈Σ+(Q,a)

Xα

)− 2i[Ad(k)H, s]

= X − 2i[Y, s] .

In the third equality we use equation (1.26) and in the last line the reparametri-sations RIII , RII and RI are understood to be undone.

Application of Stokes’ theorem

In our application of Stokes’ theorem we essentially want to use ∂DePS =Euclid−ePS. To that end, note that the boundary operator ∂ acts only onthe [0, 1]1+n−|L|× [1,∞)|L| part of the domain of definition of DePSL,c. Theboundary parts coinciding with boundaries of a+ are again of codimensiontwo and do not contribute (see also appendix A.2.3). The only nonvanish-ing contribution comes from t = 0 (ePS) and t = 1 (Euclid). Since wewant to postpone issues of convergence to the next two subsections, we use∂DePSε = Euclid − DePS(ε), where DePSε := DePS|t∈[ε,1]. The idea isto deform Euclid into PS as far as convergence of the integral allows, i.e.we apply Stokes in the following way:∫

DePS(ε)g(Q,A)dQ = −

∫DePSε

d(g(Q,A)dQ)︸︷︷︸=0

+∫Euclid

g(Q,A)dQ

=∫Euclid

g(Q,A)dQ .

Existence of the integral for ε > 0

In this subsection we show that the integral∫DePS(ε)

g(Q,A)dQ


exists for 1 ≥ ε > 0. The limit ε → 0 requires more care and is dis-cussed in the next subsection. Thus we have to evaluate the terms in theexponent of g(Q,A) in more detail. To this end we note some useful re-lations that are derived in section A.1.5 in appendix A. For Xα ∈ Q+,α,X,X ′ ∈

⊕α∈Σ+(Q,a)Q+α hold:

B(Xα, Xβ) = −B(φ(Xα), φ(Xβ)) = δα,βB(Xα, Xα) , (1.29)B(X,X ′) = −B(φ(X), φ(X ′)) , (1.30)

B(Xα ± φ(Xα), Xβ ± φ(Xβ)) = 0 , (1.31)B(Xα + φ(Xα), Xβ − φ(Xβ)) = δα,β2B(Xα, Xα) . (1.32)

For ε > 0 it suffices to discuss the B(Q,Q) term. Referring to (1.28),B(Ξ,Ξ) can be rewritten to

−B(Ξ,Ξ)

= Tr(M2) +∑

α∈Σ+(Q,a)

Tr[(Xα + (1− ε) tanh

[∑j

ξ(hj)α(Hj)]φ(Xα)

)2]=

(1.29)B(M,M) +

∑α∈Σ+(Q,a)

B(Xα, Xα)(

1− (1− ε)2 tanh2[∑

j

ξ(hj)α(Hj)]).

For hi ∈ [0, 1] this term guarantees convergence of the integral. The crossterm B(Ξ,Υ) is imaginary and therefore leads only to a phase factor.

Before we turn to B(Υ,Υ) let us make two observations. First note thatB(Ei, Ej) = 0, which can be seen from∑α∈Σj,i, 6=

eαi eαj B(Xs,α + sgn(α(Hi))φ(Xs,α), Xs,α + sgn(α(Hj))φ(Xs,α))

=(1.31)

0 , (1.33)

where we define Σj,i, 6= := Σi,6= ∩Σj, 6=. For the equality (1.33) we usesgn(α(Hi)) = sgn(α(Hj)). Second we have that

Ej − εθ(Ej) + ε∆Xs,j = (1− ε)Ej − ε∑

α∈Σ+(Q,a)

α(Hj)iφ(Xs,α)

holds. Since B(·, ·) is negative definite on on Q−, the following computationis enough to show B(Υ,Υ) > 0 for ε ∈ (0, 1] and thus convergence in the hi

directions for all Xα ∈ Q+,α:

B(− i∑

α∈Σ+(Q,a)

α(Hj)φ(Xs,α), Ei)

=(1.27)

−∑

α,β∈Σ+(Q,a)

eβi sgn(β(Hi))α(Hj)B(φ(Xs,α), φ(Xs,β))

= −∑

α∈Σ+(Q,a)

eαi |α(Hj)|B(φ(Xs,α), φ(Xs,α)) > 0 .


Thus we conclude that the integral over DePS(ε) exists for ε > 0.

Existence of limε→0

We first show for |L| > 0 that

limε→0

∫DePSL,c(ε)

g(Q,A)dQ = 0 . (1.34)

The reason for this result is a very general one. The convergence of the in-tegral at the boundary of PS is brought about by an oscillatory term alongthe boundary. In our case, integrating along the boundary lines yields es-sentially a regularised delta distribution. Our parametrisation is well suitedto show this mechanism explicitly. Using DePS(ε), we pull back dQ toa+×K×ZK(a)Q+. Choosing an appropriate set of charts and a decomposi-tion of unity it is enough for our argument to consider integrals over finitelymany subsets a+×Ui×Q+, with Ui ⊂ K/ZK(a). Here, it is important thatK/ZK(a) is compact. Then the integration is decomposed into an outerpart over a+ ×K/ZK(a), and an inner part over Q+. This is possible sincethe integrals are exponentially convergent (ε > 0), and Fubini’s theorem canbe applied. The Q+ integrations are essentially Gaussian times a polyno-mial coming from the Jacobian. We show that it is possible to perform thelimit ε → 0 after doing the inner Gaussian integrations. In the followingEinstein’s summation convention is in place. Schematically, the Gaussianintegrations over Q+ for each simplicial cone c ∈ C are of the form

IL,c(ε,H, [k]) := e−higi

∫RdimQ+

∏l

dxl e−(xl)2fl−2ixlgl P ,

where gi(ε, [k]), fl(ε, hi, [k]) and gl(ε, hi, [k]) are functions of [k] ∈ K/ZK(a), εand H = hiHi ∈ a+ to be specified later on. P is a polynomial in ε, xl, hi,[k]and ∂r tanh(ξ(hi)α(Hi)), where ∂r represents arbitrary partial derivativeswith respect to hi. Note that i = 1, . . . ,dim a and l = 1, . . . ,dimQ+. Nowit is possible to introduce source terms, and to perform the integral

IL,c(ε,H, [k]) = e−higi P ′(∂jl|jl=0

, . . . )∫dxle−fl(x

l)2−2ixl(gl+jl)

= e−higi P ′′(∂jl|jl=0

, . . . )∏l

√π

fle− (gl+jl)

2

fl

= e−higi P ′′′(

1fl, . . . )

∏l

√π

fle−g2lfl , (1.35)

where primes just indicate that these are different polynomials, and the dotsrepresent a dependence on ε, hi,[k] and ∂r tanh(ξ(hi)α(Hi)). We will show


that for ε = 0 we have f1 = 0 and g1 6= 0. Furthermore we show that fl ≥ 0and gl ∈ R for all l. Then the exponential dominates the polynomial, and(1.35) is zero for ε = 0. In addition we show that gi > 0, and hence theremaining integrals over a+ are convergent.

Thus the issue of convergence is reduced to a discussion of the functionsgi, fl and gl. gi is read off from

2iB(Υ, A) = (hi − 1 + ε)gi ,

which gives

gi = 2iθ(hi − 1)B(Ad(k)(Ei − εθ(Ei) + ε∆Xs,i), A) .

Remembering inequality (1.24) we conclude that gi > 0 for small enough ε.In the following we restrict ourselves to the Q+,α integrations, since the

integrations over Q+,0 are trivially convergent. The functions fl and gldepend on the choice of basis of Q+. For a good choice, inequality (1.24)is important. Let ΠQ+ denote the orthogonal projection onto Q+, then wechoose j ∈ L and define

X1 := iΠQ+(Ej) =∑

α∈Σj,6=

eαjXs,α (1.36)

as the first basis vector, and extend this to an orthonormal basis of⊕α(Hj,c)6=0Q+,α,which fixes the first m basis vectors. Extend this to an orthonormal basis ofQ+ that respects the root decomposition for the root spaces with α(Hj) = 0.For all Xl and Xl′ we have the equality

B(Xl + φ(Xl), Xl′ + φ(Xl′)) = 0 .

This is important as it makes B(Ξ,Ξ) proportional to δl,l′ , i.e. that theGaussian integrals are diagonal. We read off

fl =∑

α∈Σl, 6=

eαl |B(Xs,α, Xs,α)|(1− (1− ε)2 tanh2(ξ(hi)α(Hi))

).

In particular we have for l = 1:

f1 = ε(2− ε)B(X1, X1) .

Similarly it is easy to check that the g′′l defined by

−2B(Ξ,Υ) = −2ixlg′′l

vanish in the limit (ε = 0 and hl ≥ 1) or are identically zero. Therefore weneglect g′′l in the following. Now, we turn to 2iB(Q,A). B(Ξ, A) containsxl only linearly and thus we define

−2iB(Ξ, A) = −2ixlg′l ,


which gives

g′l =∑

α∈Σl,6=

eαl B(Xs,α + (1− ε) tanh(ξ(hi)α(Hi))φ(Xs,α), k−1Ak) ,

and gl = g′l + g′′l . This leads to

limε→0

g1 = B(−iEj ,Ad(k−1)A) < 0 .

Putting everything together we obtain that

limε→0

IL,c(ε,H, [k]) = 0 .

In particular∫a+,L,c

IL,c(ε,H, [k]) is a bounded function, and the limit limε→0 IL,c(ε,H, [k])is uniform. Thus the limit commutes with the outer integrals. Hence thediscussion of IL,c(ε,H, [k]) directly yields the existence of the limit.

Reaching PS

The information about IL,c(ε,H, [k]) helps us finish the proof with the fol-lowing two equations:

limε→0

∫DePS(ε)

g(Q,A)dQ = limε→0

∫Pc∈C DePS∅,c(ε)

g(Q,A)dQ

= limε→0

∫PS

g(Q,A)χε(Q)dQ .

The first equality sign holds since the contributions of all attached surfaces|L| > 0 vanish. For the last equality sign it is important to note that for∑

c∈C DePS∅,c(ε) only the ε contained in the B(Ξ,Ξ) term is important. Allother ε can be set to zero even before executing the Gaussian integrations.This procedure affects the terms B(Ξ,Υ) and B(Q,A). The last equalityholds by identifying χε′ = exp(+ ε′

4 Tr(Q− θQ)2) and ε′ = 2ε− ε2.

Remark 1.1. To obtain the theorem when g = k⊕ p is the direct sum of asemisimple and an Abelian Lie algebra, let a′ ⊕ a denote a maximal Abeliansubalgebra of p and replace a+ by a′ × a+ and H by H ′ + H everywhere inthe proof. In addition let k′ denote the semisimple part of k and replace k byk′ everywhere in the proof.

Remark 1.2. It is possible to choose different regularisation functions χε.Nevertheless, the choice made here seems natural, as it has the highest in-variance possible.

Remark 1.3. The convergence properties can be seen quite clearly in thediscussion of Ic(ε,H, [k]). The convergence is not uniform in A. To haveuniform convergence, we need As > δ for fixed δ > 0.


Remark 1.4. In applications with As ≥ 0, one has to substitute A by A+δs.For fixed δ > 0 this gives uniform convergence in A.

Remark 1.5. In the proof, we do not have to require that the extensions ofPS are nullsurfaces. However, this idea is needed as a guiding principle forfinding the extension.

1.4.4 Equivalence of SW and Euclid

Let us note that the SW domain and the validity of the corresponding hy-perbolic Hubbard-Stratonovich transformation is discussed in detail in [7].Nevertheless we give a different proof by deforming SW into Euclid. Us-ing some of the constructions of the proof for the PS transformation thisdeformation can be stated very explicitely.

First we briefly discuss convergence of the Gaussian integral over

SW : p⊕Q+ → QC

(Y,X) 7→ X − ibeY se−Y .

For X ∈ Q+ and Y ∈ p we have that B(X,X) > 0 and

B(ibeY se−Y , ibeY se−Y ) = −b2B(s, s)

is constant. Furthermore B(X, ibeY se−Y ) is purely imaginary and

−iB(−ibeY se−Y , A) = −bTr(e−2YAs) < 0

yields convergence in the p directions.To see the properties of SW more explicitely we use the reparametrisa-

tion RI and the decomposition of s to obtain:

SW RI : a+ ×K/ZK(a)×Q+ → QC

(H, [k], X) 7→X − ibAd(k)[Ms+∑

α∈Σ+(Q,a)

[cosh(α(H))Xs,α + sinh(α(H))φ(Xs,α)]].

(1.37)

Since the image of the boundary of a+ is again of codimension at least two,the parametrisation (1.37) clearly shows that ∂SW = 0.

A suitable deformation is given by

DSW : [0, 1]× a+ ×K/ZK(a)×Q+ → QC

(t,H, [k], X) 7→ X − ibAd(k)[(1− t)Ms+∑

α∈Σ+(Q,a)

[(cosh((1− t)α(H))− 1)Xs,α +sinh((1− t)α(H))

1− tφ(Xs,α)]

]


Note that DSW (t = 0) = SW and DSW (1, H, [k], X) = X + i[kHk−1, s].Since [p, s] = Q− we obtain DSW (1) = Euclid.

To complete the argument we show that the integral over DSW is con-vergent. Therefore note that

B(Q,Q) =B(X,X)︸︷︷︸>0

+∑

α∈Σ+(Q,a)

(2t− t2)sinh2((1− t)α(H))

(1− t)2B(Xs,α, Xs,α)︸︷︷︸

>0

+ . . . ,

where the dots represent unimportant terms. These are terms which arepurely imaginary, terms which are linear in sinh and all terms containingMs. Thus we have convergence for t > 0. For t = 0 convergence is generatedby the B(Q,A) term, as discussed above for the SW parametrisation.

1.4.5 Different representations of the integral

In this section we discuss different possibilities to represent the integral overthe PS domain. In particular we want to derive corollary 1.1 and 1.2.

Assume that k⊕Q+ is the direct sum of an Abelian and a semisimple Liealgebra. Then choose a maximal Abelian subalgebra h of Q+ and let h =h′ ⊕ a be the corresponding decomposition into the Abelian and semisimplepart. Then we have the following reparametrisation:

R : p×K/ZK(a)× ao+ × h′ → p⊕Q+

(Y, [k], H, J ′) 7→ Y + k(H +H ′)k−1 .

If g is semisimple we can use Cartan decomposition to define a semisimpleLie group as G := epK. This yields the reparametrisation

R : G/ZK(a)× ao+ × h′ → p⊕Q+

(eY [k], H,H ′) 7→ Y + k(H +H ′)k−1 .

The parametrisation of the PS domain which is most frequently used inthe literature, is

PS R : G/ZK(a)× ao+ × h′ → Q([g], H,H ′) 7→ g(H +H ′)g−1 .

In the following we present a derivation of corollary 1.1. This is done in adetailed way that clearly exhibits the origin and form of J ′ in 1.1. In sectionA.2.2 we show that the pullback of dQ by PS R is given by

(PS R)∗dQ = ∆(H +H ′) dµ([g]) ∧ dH ,


where dµ([g]) is a left invariant volume form on G/ZK(a) and ∆ is given by

∆(H +H ′) =∏


α(H +H ′)dα ·∏

α∈Σ+(k⊕Q+,h)

α(H)dα . (1.38)

This needs further explanation: dα denotes the dimension of the weightspace corresponding to α. The weights α(H +H ′) are real since [H +H ′, ·]is hermitian with respect to Tr(XY †). Furthermore α(H ′) = 0 for α ∈Σ+(k⊕Q+, h).

It is important that ∆ differs from J ′ in corollary 1.1 only by taking themodulus of the weights in Σ+(k⊕Q+, h). But the roots α ∈ Σ+(k⊕Q+, h)are positive when evaluated on ao+. Therefore we have the following equality:∫

PSRf(Q) dQ =

∫idf(g(H +H ′)g−1) · J ′(H +H ′) dµ([g]) ∧ dH ,

where id denotes the identity on G/ZK(a) × ao+ × h′. Now it is possible toreplace the volume form dµ([g]) by the left invariant measure |dµ([g])| anddH by Lebesgue measure |dH| on h:∫

PSR

f(Q) dQ =∫

G/ZK(a)×ao+×h′

f(g(H +H ′)g−1) · J ′(H +H ′) |dµ([g])||dH| .

Replacing G/ZK(a) by G introduces only a constant factor c′ ∈ R \ 0:c′ ∈ R \ 0 :∫

PSR

f(Q) dQ = c′∫

G×ao+×h′

f(g(H +H ′)g−1) · J ′(H +H ′) |dµ(g)||dH| ,

where |dµ(g)| denotes the invariant measure on G. Since now |dµ(g)| is inparticular right invariant we can use that the action of the Weyl group ona+ generates a. To exploit this property we need that J ′ is also invariantunder the action of the Weyl group. Recall that J ′ is given by

J ′(λ) =∏


α(λ)dα∏

α∈Σ+(k⊕Q+,h)

|α(λ)dα | .

The second factor is trivially invariant, whereas for the first factor an addi-tional argument is needed. Therefore note that we used s to define a notionof positivity for the weights α ∈ Σ+(p⊕Q−, h). Since s is Ad(K) invariantwe conclude that the action of the Weyl group only permutes the weights inΣ+(p⊕Q−, h) and hence it is invariant. Thus we have∫

PSR

f(Q) dQ = c′′∫

G×h

f(g(H +H ′)g−1) · J ′(H +H ′) |dµ(g)||dH| ,

1.5. EXAMPLES 33

where c′′ ∈ R0 is a constant. Setting f = g · χε we obtain corollary 1.1.If PS is nearly everywhere injective and regular, then so is PS R. This

allows application of the change of variable theorem, which yields corollary1.2.

1.5 Examples

1.5.1 U(p, q) symmetry

This case has been proven by Fyodorov [10] using different methods. Thegeneral theorem (1.2) can by applied by choosing gl(p + q,C) as complexLie algebra and defining s = Diag(1q,−1p). No additional involutions τi areneeded. In this setting, the maximal Abelian subalgebra h ⊂ Q+ is givenby the real diagonal matrices. In addition we have

k⊕Q+ = x ∈ gl(n,C)|X = sXs and p⊕Q− = x ∈ gl(n,C)|X = −sXs

In the following λ := Diag(λ1, . . . , λp+q) ∈ h denotes a real diagonal matrix.The weights are given by fi − fj where i 6= j and fi(λ) = λi. The corre-sponding weight spaces are given by the matrix Eij , which is nonzero onlyin the i, jth entry. Thus every weight has a two dimensional weight space,i.e., complex one dimensional. Thus we have

J ′(λ) =∏i<j

|λi − λj |2 .

1.5.2 O(p, q) symmetry

This case has been proven by Fyodorov, Wei and Zirnbauer [12]. In addi-tion to the involutions of the pseudounitary setting we have an involutionτ1(X) = −sXts and σ1 = −1. The additional involution just requires allmatrices to be real. Therefore the weight spaces are now one dimensional,and give rise to non trivial signs:

J ′(λ) =∏


α(λ)∏

α∈Σ+(k⊕Q+,h)

|α(λ)|

=∏

i≤p<j≤p+q(λii − λjj)

∏i<j≤p,p<i<j≤p+q

|λi − λj |

=∏i<j

|λi − λj |p∏i=1

p+q∏j=p+1

sgn(λi − λj) ,

which is precisely corollary one in [12].


1.5.3 Two dimensional case

The two dimensional example considered at the beginning of the chaptercan also be put into the general framework. Consider the O(1, 1) case,and require all matrices to be traceless. This means exchanging gl(2,C) bysl(2,C). Then all relevant spaces are:

Q+ = R(

1 00 −1

), Q− = R

(0 1−1 0

),

p = R(

0 11 0

), k = 0 .

Hence there are only two roots ±(f1 − f2) ∈ Σ(p⊕Q−, h), and J ′ containsonly one factor. But this factor is vital since it yields a nontrivial sign.Identifying

e1 =1√2

(1 00 −1

)and e2 =

1√2

(0 1−1 0

),

we obtain that B(ei, ej) = Tr(eiej) = (−1)i+1δij .

1.5.4 Simple system of interacting bosons

In the following we sketch a simple application of a hyperbolic Hubbard-Stratonovich transformation in the context of interacting bosons. Let a†

and a denote bosonic creation and annihilation operators and define chargeand pair annihilation operators:

Q :=N∑i=1

a†iai and P :=N∑i=1

aiai .

Then we consider the simple interaction Hamiltonian given by

Hint = eQ2 − bP †P ,

which has to satisfy the stability condition e > b. Using boson coherentstates, and neglecting normal ordering terms for the moment, the Hamilto-nian is given by

Hint(z, z) = e

(N∑i=1

zizi

)− b

(N∑i=1

zizi

)(N∑i=1

zizi

).

Introducing the matrix

A(z, z) :=1√2

(√e∑N

i=1 zizi√b∑N

i=1 zizi√b∑N

i=1 zizi −√e∑N

i=1 zizi

)

1.5. EXAMPLES 35

the interaction Hamiltonian is equal to:

Hint(z, z) = TrA(z, z)2 .

The stability condition translates to As > 0, with s = Diag(1,−1). Fur-thermore, A satisfies the symmetry relations A = sA†s and A = −ΩtAtΩ,with

Ω =(

0 1−1 0

).

Let us now state how this fits into the general setting of theorem 1.2: Choosesl(2,C) as complex Lie algebra and the additional involution τ1(X) = −ΩtXtΩ,with σ1 = 1. The relevant spaces are

k = iR(

1 00 −1

), p = R

(0 11 0

)+ iR

(0 1−1 0

),

Q+ = R(

1 00 −1

), Q− = R

(0 1−1 0

)+ iR

(0 11 0

).

Now it is possible to apply theorem 1.2 to decouple the interaction term inthe boson coherent state path integral representation of the partition func-tion of the system. Note however that we have to make the usual continuumapproximation that z(t) is complex conjugate to z(t).13 After the applica-tion of theorem 1.2 the z and z integrations can be performed. We do notproceed any further in this direction, since the intention was only to illus-trate a possible application to many body systems with bosonic degrees offreedom.

13This is an approximation since z(t) and z(t) stem from neighbouring time steps in thediscrete time path integral.

Chapter 2

Bosonisation of granularfermionic systems

In this chapter we derive a bosonic path integral representation of the grandcanonical partition function of a granular fermionic system.1 In the followingwe give an overview of the organisation of this chapter.

In section 2.2 we give a derivation of the bosonic path integral represen-tation of the grand canonical partition function of a granular fermionic sys-tem starting from a Grassmann coherent state path integral representationof the grand canonical partition function. Then colour-flavour transforma-tion is applied iteratively for each time step, which leads to a factorisationof the Grassmann integrals. Integrating out the Grassmann variables leadsto the bosonic path integral representation. In section 2.3 we give anotherderivation based on a suitably enlarged Fock space. In this larger Fock spacegeneralised coherent states can be used to obtain the result. Finally in sec-tion 2.4 we calculate the contribution of fluctuations in the semiclassicallimit. Essentially this amounts to calculate the fluctuation determinant forgeneralised coherent state path integrals. To the best of our knowledge thereexists no derivation for the general case. We do the calculation in discretetime using a method invented by Forman [32].

2.1 Granular bosonisation via colour-flavour trans-formation

In the first subsection we fix the setting and discuss three different symme-try classes. Furthermore we state a colour-flavour transformation for eachsymmetry class. This transformation is used in the next section to decou-ple the time evolution within the Grassmann coherent state path integral,

1For a detailed description of the context and the motivation for this chapter see thesecond part of the introduction.

37

38 CHAPTER 2. BOSONISATION OF GRANULAR SYSTEMS

i.e. to convert the ‘normalisation factors’ exp(−∑

k ψkψk) into expressionscontaining only pairs ψkψk,ψkψk and ψkψk−1, that are invariants of the cor-responding group.2 In the third subsection we integrate out the Grassmannvariables. This leads to an effective action which allows a formal continuumlimit.

2.1.1 Setting and symmetry classes

Our starting point is the discrete time Grassmann functional integral repre-sentation of the grand canonical partition function [35]:

Tr(exp(−βH)) = limM→∞

∫DM (ψ, ψ) exp(−SM [ψ, ψ]) , (2.1)

where H denotes the Hamiltonian of the granular system. Note that weabsorb the chemical potential into the Hamiltonian to simplify notation.The discrete time action SM is given by

−SM [ψ, ψ] = −M−1∑k=0

ψkψk +M∑k=1

ψkψk−1 −β

M

M∑k=1

H(ψk, ψk−1

), (2.2)

with antiperiodic boundary conditions ψM = −ψ0. Let us first hide the timestep index. The remaining index structure is given by

ψ = (ψ11, ..., ψ

iα, ..., ψ

N ′Ne)

ψ = (ψ11, ..., ψ

iα, ..., ψ

N ′Ne)

t .

The upper indices are called internal and the lower external. To distinguishbetween external and time indices the former are denoted by greek and thelatter by latin letters. If the internal or external indices are not explicitlyshown, they are summed over. Internal indices will not be denoted by t sothere is no confusion with transposition.

The Hamiltonian is required to have a unitary, orthogonal or unitarysymplectic symmetry. This has the consequence that the Hamiltonian is apolynomial in the generators of the Howe dual group (see appendix B.2.4and theorem B.1). The generators are exactly the bilinear invariants of thesymmetry group K ∈ U(N ′),O(N ′),USp(N ′). See table 2.1 for a list offermionic Howe dual pairs (K,G). Let k ∈ K, then the group action is givenby

ψj 7→ ψjk−1 ≡ ψj(1Ne ⊗ k−1) =

N ′∑i=1

ψij,αk−1im

ψl 7→ kψl ≡ (1Ne ⊗ k)ψl =N ′∑i=1

kmiψil,α , (2.3)

2Here the index structure has been suppressed deliberately. It will be presented indetail in the next section.

2.1. GRANULAR BOSONISATION VIA CFT 39

SO(2N) K G H

N = (p+ q)Ni U(Ni) U(p+ q) U(p)×U(q)N = NeNi O(Ni) SO(2Ne) U(Ne)N = 2NeNi USp(2Ni) USp(2Ne) U(Ne)

Table 2.1: (K,G) denotes a Fermionic Howe dual pair.

with time indices j and l. In the following we discuss the situation for eachsymmetry group. In the case of unitary symmetry we set N ′ = Ni. Thequadratic invariants of U(Ni) are given by

ψαψβ =Ni∑i=1

ψiαψiβ .

This corresponds to the special case of unitary Howe dual pairs where p = Ne

and q = 0 (see table 2.1). We restrict ourselves to this case since we are notaware of a physical system that requires an arbitrary p and q. However thefollowing constructions readily generalise to the case of arbitrary p and q.For orthogonal symmetry we set N ′ = Ni and the invariants are given by

ψαψβ =Ni∑i=1

ψiαψiβ , ψ

tαψβ =

Ni∑i=1

ψiαψiβ and ψtαψβ =

Ni∑i=1

ψiαψiβ .

Moreover, for unitary symplectic symmetry we set N ′ = 2Ni. The upperindex i is seen as a composite index (i, s), with s = ±1. The invariants aregiven by

ψαψβ =Ni∑i=1

∑s=±1

ψi,sα ψi,sβ , ψtαJ

′ψβ =Ni∑i=1

∑s=±1

ψi,sα sψi,−sβ

and ψtαJ′ψβ =

Ni∑i=1

∑s=±1

ψi,sα sψi,−sβ ,

and J ′ is the symplectic unit acting on the upper indices. Note that fork ∈ USp(2Ni) we have ktJ ′ = J ′k−1.

We need some preparations to be able to state the colour-flavour trans-formations, which we want to use in the next section. First we define

Ψ :=(ψj ψtl

)Ψ :=

(ψjψtl

)for fixed time indices j and l. The colour-flavour transformations for thethree different symmetry classes are derived in appendix B.2.4. In the fol-lowing we just state without derivation all objects that are needed to define


the colour-flavour transformation. In our application of the colour-flavourtransformation we consider two neighbouring time steps. For this reason wehave take twice as many degrees of freedom into account in the applicationof the colour-flavour transformation. Therefore we use the colour-flavourtransformation corresponding to the groups summarised in table 2.2 insteadof the ones in table 2.1. In addition we need a complex vector space W

SO(2N) K G H

N = 2NeNi U(Ni) U(2Ne) U(Ne)×U(Ne)N = 2NeNi O(Ni) SO(4Ne) U(2Ne)N = 4NeNi USp(2Ni) USp(4Ne) U(2Ne)

Table 2.2: List of Howe dual pairs needed for the colour-flavour transforma-tions we use.

and the measure dµ(Z†e , Ze) defined in table 2.3. W gives a parametrisation

K W f(Z†e , Ze)U(Ni) Ze ∈W2Ne |sZes = −Ze det−Ne(1 + Z†eZe)O(Ni) W2Ne det−2Ne(1 + Z†eZe)

USp(2Ni) Ze ∈W2Ne |sZes = −Ze, JZeJ t = Ze det−Ne(1 + Z†eZe)

Table 2.3: Wn := Ze ∈ End(Cn)|Zte = −Ze, s = 1Ne ⊗ σz and J =i1Ne ⊗ σy. W parametrises the coset space G/H. Let dw denote Lebesguemeasure on W then we define dµ(Z†e , Ze) := f(Z†e , Ze)dw.

of the coset space G/H, and dµ(Z†e , Ze) denotes the left invariant measure.Now, we can state the colour-flavour transformation:∫

Kdk exp

[Ψ(k 00 k−1t

)Ψ]

=∫Wdµ(Z†e , Ze) det−Ni/2(1 + Z†eZe) exp

[12

ΨZΨt +12

ΨtZ†Ψ],

(2.4)

where Z := Ze ⊗ 1Ni and Ze ∈ W . We need for the derivation of granularbosonisation that Z commutes with k.

2.1.2 Application of the colour-flavour transformation

In this section we promote the global symmetry (2.3) of the Hamiltonianto a local one, which leads to a factorisation of the Grassmann integrals.We proceed iteratively, i.e., for one time step after the other. First we


introduce an average over the symmetry group, and then we apply colour-flavour transformation.

First step: We perform transformation (2.3) for j = 1 and l = 0 in thediscrete time path integral representation and average over K. The corre-sponding integration variable is called k1. Only the first two summands inthe first term in (2.2) are affected by this transformation. The two sum-mands are given by

−ψ1k−11 ψ1 − ψ0k1ψ0 =

(−ψ0 ψt1

)(k1 00 k−1

1t

)(ψ0

ψt1

).

The right hand side above is exactly the left hand side of the colour-flavourtransformation (2.4). Applying (2.4) yields

12(−ψ0 ψt1

)Z1

(−ψt0ψ1

)+

12(ψt0 ψ1

)Z†1

(ψ0

ψt1

). (2.5)

Next we use the well known identity

f(ξ) =∫

dηdη e−ηη+ξηf(η) (2.6)

to change the term ψ1ψ0 to −η1η1 + ψ1η1 + η1ψ0. Adding this to (2.5) weobtain

12(−ψ0 ψt1

)Z1

(−ψt0ψ1

)+

12(η1 ηt1 ψt0 ψ1

)(−J σ3

−σ3 Z†1

)ηt1η1

ψ0

ψt1

. (2.7)

The transformation η1 7→ −η1 brings the second term to a more convenientform. In the last step we will make a similar transformation that compen-sates signs factors which might arise from the first transformation. The firstterm in (2.7) is dealt with in the next step.

Second step: Now we do the transformation (2.3) for j = 2 and l =1 and average over K with integration variable k2. This transformationchanges the first term in (2.7) and the term −ψ2ψ2 in (2.2) into

12(−ψ0 (k2ψ1)t

)Z1

(−ψt0k2ψ1

)− ψ2k

−12 ψ2

=12(−ψ0k2 ψt1

)Z1

(−kt2ψt0ψ1

)− ψ2k

−12 ψ2 .

Using (2.6) to decouple −ψ0k2 gives

−η2η2 − ψ0k2η2 +12(η2 ψt1

)Z1

(ηt2ψ1

)− ψ2k

−12 ψ2

= −η2η2 +(−ψ0 ψt2

)(k2 00 k−1

2t

)(η2

ψt2

)+

12(η2 ψt1

)Z1

(ηt2ψ1

)


Now colour-flavour transformation can be applied:

12(ηt2 ψ2

)Z†2

(η2

ψt2

)+

12(η2 ψt1

)Z1

(ηt2ψ1

)− η2η2

+12(−ψ0 ψt2

)Z2

(−ψt0ψ2

).

The term in the second line will be treated in step three. Combining theother terms with ψ2ψ1 we obtain

12(η2 ψt1 ηt2 ψ2

)(Z1 −11 Z†2

)ηt2ψ1

η2

ψt2

.

Third step: Now we have nearly the same starting point as in step two,and we can iterate the procedure until step M − 1.

Last (Mth) step: Our starting point is

12(−ψ0 ψtM−1

)ZM−1

(−ψt0ψM−1

)+ ψMψM−1

=12(ψM ψtM−1

)ZM−1

(ψMψM−1

)+ ψtMψM−1 ,

where we have used that ψM = −ψ0. These are the antiperiodic boundaryconditions that arise in the Grassmann coherent state representation of thetrace. Application of identity (2.6) yields

12(ψM ψtM−1

)ZM−1

(ψtMψM−1

)− ηMηM + ψMηM + ηMψM−1 ,

which can be written as

12(ψM ψtM−1 ηtM ηM

)(ZM−1 σ3

−σ3 J

)ψtMψtM−1

ηMηtM

.

As in the first step, we can make an additional transformation ηM 7→ −ηM .Thus we have obtained a path integral representation which allows a fac-torisation of the Grassmann integrals. We have introduced new Grassmannvariables η and η and new bosonic variables Z and Z†. Note that the pathintegral representation is local in discrete time. However at this stage it isnot clear how to perform the (formal) continuum limit M → ∞. Obtain-ing such a continuum limit is the aim of the next section. To simplify thenotation we set Z0 = J and Z†M = J† = −J .


2.1.3 Effective Hamiltonian and the continuum limit

In the last section the time evolution of the Grassmann variables was de-coupled at the expense of introducing new fields. In this section we wantto integrate out the Grassmann variables to obtain an effective Hamiltonianin the new fields. Therefore we define the mean value of a function F (ξ) ofGrassmann variables ξ as

〈F 〉X =∫

dξe12ξtXξF (ξ)∫

dξe12ξtXξ

.

In this situation we have the following Wick theorem for Grassmann vari-ables:

〈exp(χξ)〉X = exp(−1

2χtX−1χ

).

Let us first define

Xk :=(Zk−1 −11 Z†k

)as an important building block. To this end note the basic Gaussian integral∫

dξ e12ξtXkξ = det1/2(1+ Z†kZk−1) ,

and the useful identity

X−1k =

(Zk−1 −11 Z†k

)−1

=

((1+ Z†kZk−1)−1Z†k (1+ Z†kZk−1)−1

−(1+ Zk−1Z†k)−1 Zk−1(1+ Z†kZk−1)−1

).

Now, integrating out the Grassmann variables yields∫DMµ(Z†, Z)

det1/2(1+ Z†1J) det1/2(1+ J†ZM−1)

det1/2(1+ Z†1Z1)M−1∏k=2

det1/2(1+ Z†kZk−1)

det1/2(1+ Z†kZk)

(1− β

M〈H〉Xk

),

where we defined

DMµ(Z†, Z) :=M−1∏k=1

dµ(Z†k, Zk) .

Recalling Z†M = J† and Z0 = J , we obtain

Tr exp(−βH) = limM→∞

∫DMµ(Z†, Z) exp(−SM [Z†, Z])


with

−SM [Z†, Z] =12

tr ln(1 + Z†MZM−1) +12

M−1∑k=1

tr ln1 + Z†kZk−1

1+ Z†kZk

− β

M

∑k

〈H〉Xk .

This expression allows for a formal continuum limit. Defining

H(Z†, Z) := 〈H〉“Z −11 Z†

”the continuum limit of SM [Z†, Z] can be identified as

−S[Z†, Z] =Ni

2tr(ln(1+ Z†(β)Z(β))

− Ni

2

∫ β

0dτ tr(Z†∂τZ(1+ Z†Z))−

∫ β

0dτH(Z†, Z) . (2.8)

The boundary conditions are Z(0) = J and Z† = J†. Variation of (2.8)leads to the following saddle point equations

∂τZ† =

2Ni

(1+ Z†Z)(∂ZtH)(1+ ZZ†)

∂τZ = − 2Ni

(1 + ZZ†)(∂ZH)(1+ Z†Z) .

Let us summarise what we have achieved so far: We have obtained a pathintegral representation of the grand canonical partition function in termsof purely bosonic variables. We use the word ‘granular bosonisation’ forthis way of representing a grand canonical partition function of a granularfermionic system. The representation closely resembles generalised coherentstate path integrals [35] with rather strange boundary conditions and aclassical limit that is controlled by Ni.

In the next section the connection to generalised coherent state pathintegral is made precise. This connection sheds some light on the origin andphysical meaning of the boundary conditions. Furthermore we make contactwith the large body of literature concerning coherent states and semiclassicallimits.

2.2 Fock space approach to granular bosonisation

In the previous section we obtained a path integral description in bosonicvariables of the grand canonical partition function of a granular fermionicsystem. In particular, the representation suggests that the large Ni limit is

2.2. FOCK SPACE APPROACH TO GRANULAR BOSONISATION 45

a classical limit, and that the classical phase space is G/H. However, it isnot so clear how to interpret a classical state Z and the boundary conditionsin terms of the original system. The aim of this section is to propose ananswer to these questions. In addition, this leads to a different derivationof the path integral representation.

Let us first give a rough motivation for the course of action we take: Thepath integral representation of the last section is very similar to well knowngeneralised coherent state path integrals. Such coherent states are used inthe derivation of colour-flavour transformation to represent the projector Pin appendix B.2.4. These coherent states come with a Fock space containingtwice as many fermions as the original system. This Fock space can be seenas the tensor product of two copies of the original Fock space of our system.We will take the view that one is the state space of our system and the otheris an ancilla system. The details are discussed in subsection 2.2.1. Granularbosonisation is derived and discussed in subsection 2.2.2.

2.2.1 Coherent states and a semiclassical limit

The Fock space of system and ancilla system is generated by fermioniccreation and annihilation operators ciα

† and ciα, with α = 1, . . . , 2Ne andi = 1, . . . , N ′. Let |0〉 denote the vacuum state. The state space of the sys-tem HS is generated by creation operators with α ≤ Ne, and the state spaceHA of the ancilla system is generated by creation operators with α > Ne.The coherent states are defined as

|Z〉 := exp

∑α,β,i

ciα†Zαβc

iβ†

|0〉 .Z is an element of the complex vector space W defined in table 2.3 for eachsymmetry class. In addition we know that

P =∫W

dµ(Z†, Z)〈Z|Z〉

|Z〉〈Z|

projects onto the space of states I0 which are invariant with respect to thesymmetry group K. In the following we will work only with the subspace I0.It is well known that in this setting the limit of large Ni is a classical limit[33, 34]. This implies that expectation values factorise, and the Schrodingerequation becomes Hamiltons equation of motion. The phase space is G/H,which comes with a natural symplectic structure. The heuristic way to seethis is to use P as a representation of unity on I0 to obtain a path integralrepresentation. Let us stress again that this representation of P is onlypossible since I0 carries an irreducible representation of G. This followsfrom the fact that we consider Howe dual pairs (K,G). Next we define the


reduced density matrix of pure states |Z〉:

ρ(Z†, Z) = TrHA

(|Z〉〈Z|〈Z|Z〉

).

The expectation value of an observable As of the system in state ρ(Z†, Z)is given by

TrHS (Asρ(Z†, Z)) =Tr(As ⊗ 1HA |Z〉〈Z|)

〈Z|Z〉=〈Z|As ⊗ 1HA |Z〉

〈Z|Z〉.

2.2.2 Derivation of granular bosonisation

The key to the derivation of granular bosonisation is hidden in the bound-ary conditions of the path integral representation in the last section. Theboundary conditions are Z(0) = J and Z†(β) = J†. This motivates a furtherinvestigation of |J〉. An easy calculation shows that

ρ(J†, J) =1HS

dimHS.

Thus, in the state |J〉 system and ancilla system are entangled in such a waythat the reduced density gives uniform weight to all states. Such a stateis called maximally entangled state. Now, the grand canonical partitionfunction is given by

TrS(e−βH) = dimHS TrS(e−βHρ(J†, J)) = 〈J |e−βH ⊗ 1HA |J〉 .

Since H has K as a symmetry group its action leaves I0 invariant. Thereforewe can use P as a resolution of unity to derive a path integral representation.

To interpret the states Z in the path integral let us first summarisethe results of the above. The combined system has a well defined classicallimit and its phase space is given by the coset space G/H. For each pointZ in phase space there exists a coherent state |Z〉. The state |Z〉 can beinterpreted in terms of the original system as a reduced density matrixρ(Z†, Z). The boundary condition Z(0) = J corresponds to a state of thejoint system whose reduced density matrix is the state of maximal entropyfor the system.

2.3 Contributions of fluctuations

In this section we calculate the contributions of fluctuations to the semiclas-sical limit of the partition function. In essence this implies the calculationof a fluctuation determinant. Although a rather general result by Kochetov[40] can be applied to our case, we make some effort to rederive it from thediscrete time definition of a generalised coherent state path integral. The

2.3. CONTRIBUTIONS OF FLUCTUATIONS 47

reasons for this approach are as follows: First, to the best of our knowledgeno derivation of the result is available. Only the special case of the spin pathintegral is treated in [40, 41]. And even this derivation had to be improvedin [42]. Second, there are subtleties involved in the semiclassical limit ofgeneralised coherent state path integrals which are not well known. Third,the method we use seems to be relatively unknown and deserves a detailedexposition.

First we explain some of the subtleties to be dealt with and point to somereferences. After that we state the precise setting and the result. Finallywe use Forman’s method to derive the contributions of fluctuations.

2.3.1 Subtleties concerning the evaluation of coherent statepath integrals

Calculating the contribution of fluctuations is in principle straightforward:After determining the saddlepoint, i.e., the solution of the classical equa-tions of motion, we calculate the second variation of the action, perform thecorresponding gaussian integral, and finally compute the determinant whichis produced.

In the case of generalised coherent state path integrals there are someobstacles to this scheme. First of all the quantum problem imposes bound-ary conditions for Z(0) and Z†(T ). But a classical solution is already fullydetermined by only one of these conditions. Two solutions have been pro-posed to the ‘problem of overspecification’. The first, suggested by Faddeev[38], is to take Z and Z† as independent variables. The second solution,proposed by Klauder [37], consists of adding a small second order derivativeterm. We follow the first suggestion. This means that even in real time, thesolutions Z(t) and Z†(t) of the classical equations of motions are in generalnot complex conjugate to each other. These solutions can be reached via acontour deformation into a larger (complexified) space, which is obtained bymaking Z and Z† independent. This leads to a problem when calculatingthe second variation, since the variation should take place in some surfacewithin the enlarged space. When this issue is ignored the result turns outto be correct only up to a phase factor.

Let us first give a short review of the relevant results for the spin pathintegral. Note that for the spin path integral Z is just a complex number.Solari [39] calculated the fluctuation determinant for the spin path integralusing the discrete time action. He discovered an additional phase factorwhich had been previously overlooked. A few years later this factor wasrediscovered by Kochetov [41]. He calculated the fluctuations for the spinpath integral from the continuous action. In [40] Kochetov suggests a resultfor arbitrary generalised spin coherent states. His derivation for the spinpath integral was extended and improved by Stone et al. [42].

For the path integral of a canonical degree of freedom the situation is


similar. We refer to an exhaustive discussion by Baranger et al. [43], wherethe fluctuation determinant is calculated in discrete time.

There are only very few discussions of the contribution of fluctuations inthe case of more general coherent state path integrals. As noted above thereis the statement of a result without proof by Kochetov [40]. In addition thereis work by Ribeiro et al. [44]. They derive the fluctuation determinant for aspin coupled to a canonical degree of freedom in discrete time. Their methodis rather cumbersome and not tailored for higher dimensional generalisation.In fact their result is a special case of the result by Kochetov.

The situation can be summarised as follows: for a spin and a canonicaldegree of freedom the contributions of fluctuations is well understood. Nev-ertheless for the case of arbitrary generalised coherent states a clear deriva-tion confirming the result of Kochetov [40] is still lacking. In the followingwe provide such a derivation using context and notation of the discussion ofgranular bosonisation.

2.3.2 Setting and result

The generalised coherent states we used in the Fock space approach to gran-ular bosonisation are points in a Kahler manifolds. See for example thereview [26] for an overview of the topic. For our purposes it is not necessaryto elaborate on these structures. However the objects we introduce and theway we perform the calculations are motivated by the available structure.

To make the notation easier we define

F (Z, Z) := det(1 + Z†Z) . (2.9)

Note that lnF is called the Kahler potential. Furthermore we introducecomposite indices α = (a1, a2) such that Zα := Za1a2 . In the followingwe use Einstein summation convention, summation over a composite indeximplies summation over its two subindices. Nevertheless, k still denotes thetime steps, whereas the greek indices correspond to the composite indices.The metric is given by

gαβ := ∂Zα∂Zβ lnF .

Here we have to distinguish between indices belonging to Z or to Z. Indicesbelonging to Z are written with a bar whenever confusion might otherwisearise. Then, an easy calculation confirms that the volume element is givenby the determinant of the metric as it should. The inverse of the metric gαγ

is defined by gαβgαγ = δγ

β. Using (2.9) the continuous action (2.8) can be

rewritten as

−S =Ni

4[lnF (Z(β), Z(β)) + lnF (Z(0), Z(0))

]+Ni

4

∫ β

0dτ(∂Zβ lnF ∂τ Zβ − ∂Zα lnF ∂τZα

)−∫ β

0dτH .


In the following we state what the contributions of fluctuations look like.We derive the result in the next subsection using the discrete time represen-tation.

Theorem 2.1. The fluctutation determinant with respect to (2.8) is givenby

c ·[det(∂Z(0)∂Z(β)2S/Ni)

g1/2(β)g1/2(0)

]1/2

e1

2Ni

R β0 dτ [∂Zβ

“gβγ∂ZγH

”+∂Zβ (gγβ∂ZγH)]

,

where c ∈ C is a proportionality constant that can be fixed by setting β = 0.

Remark 2.1. The unusual exponential factor is called Solari-Kochetov phasein the literature (although, in general, it is not a phase factor).

2.3.3 Derivation using Forman’s method

To be sure that the result does not depend on a specific regularisation, wecalculate the fluctuation determinant using the discretisation given by theproblem. Our result confirms a suggestion by Kochetov [40]. We use For-man’s method to calculate the determinant. Using the notation introducedin the last subsection, the action is given by

−S =Ni

2

M−1∑k=1

lnF (Zk, Zk−1)F (Zk, Zk)

+Ni

2lnF (ZM , ZM−1)− β

M

M∑k=1

H(Zk, Zk−1) .

In the following we use the notation Fk,l = F (Zk, Zl) andHk := H(Zk, Zk−1).Hence, the second derivatives evaluate to

Ak,αβ = −∂Zk,α∂Zk,β2S/Ni = ∂Zk,α∂Zk,β

(lnFk+1,k

Fk,k− 2βNiM

Hk+1

)Bk,αβ = −∂Zk,α∂Zk,β2S/Ni = ∂Zk,α∂Zk,β

(lnFk,k−1

Fk,k− 2βNiM

Hk

)Ck,αβ = −∂Zk,α∂Zk,β2S/Ni = ∂Zk,α∂Zk,β (−1) lnFk,k = −gαβ

Dk,αβ = −∂Zk−1,α∂Zk,β2S/Ni = ∂Zk−1,α

∂Zk,β

(lnFk,k−1 −

2βNiM

Hk

).

In particular, the Hessian of 2S/Ni is given by

Q :=

AM−1 −gM−1

−gM−1 BM−1 DM−1

DM−1 AM−2 −gM−2

. . .D2 A1 −g1

−g1 B1

.


Integrating the fluctuations gives a contribution proportional to Det−1/2Q.Next we combine the volume factor coming from the measure with Det−1/2Q.Therefore we define

gk,αβ := ∂Zk,α∂Zk,β lnF (Zk, Zk) .

For the kth time step the volume factor is given by det(gk,αβ). Defining theblockdiagonal matrix

R := BlockDiag(−gM−1, gM−1, . . . ,−g1, g1) , (2.10)

the contribution from the measure can be written as Det1/2R. This leadsto the definition of LA := R−1Q. Setting A′k := g−1

k Ak and similarly for B,C and D we have

LA =

−A′M−1 1−1 B′M−1 D′M−1

−D′M−1 −A′M−2 1. . .−D′2 −A′1 1

−1 B′1

.

In addition, we define the matrix

L :=

−D′M −A′M−1 1−1 B′M−1 D′M−1

−D′M−1 −A′M−2 1. . .−D′2 −A′1 1

−1 B′1 D′0

.

Next, we view L and LA as matrix representations of finite difference opera-tors on the vectorspace V of functions that live on the links of the time lat-tice. The set of links is indexed by N := 1, 3, . . . ,M, and the vector spaceof functions is defined as V := f : N → C2N2

e . Furthermore, let eα, eαdenote an orthonormal basis of CN2

e ×CN2e with respect to a scalar product

〈·, ·〉. We define a basis fZα,i, fZα,i of V which satisfies fZα,i(j) = eαδi,j andfZα,i(j) = eαδi,j . Demanding that this basis is orthonormal defines a scalarproduct on V . Now, we view L in the obvious way as the matrix represen-tation of a linear operator on V with respect to the given basis. In additionwe can define the subspace A = f ∈ V |〈eα, f(M)〉 = 0 = 〈eα, f(1)〉. LetπA : V → A be the orthogonal projection onto A. Then, it can be easilychecked that LA = πALπA. Thus we see that the matrix LA, of which wewant to calculate the determinant, corresponds to a finite difference operatoron V with certain boundary conditions imposed.


Forman’s theorem

Forman’s theorem [32] can be formulated as follows:

Theorem 2.2. Let V be a finite dimensional Euclidean or Hermitian vectorspace and let L : V → V denote a linear mapping with kernel K ⊂ V .Moreover, let C ⊂ V be a subspace of V with the property V = C ⊕ K.Define a restriction of L via LA := πALπA : A → A where A ⊂ V , and πAis the orthogonal projection onto A. Let ρCK : V → K, v 7→ ρCK(v) denote theprojection operator onto K with respect to the decomposition V = C ⊕ K.Furthermore A⊥ denotes the orthogonal complement of A within V . Finally,vol(V ) = vol(A⊥)∧vol(A) = vol(C)∧vol(C⊥) denote the volume forms thatare used to calculate the determinants. In this situation one has:

DetLA = det(πA⊥ρCK : C⊥ → A⊥)×Det(L : C → Image(L)) Det(πA : Image(L)→ A) .

(2.11)

The theorem is useful if DetL : C → Image(L) and the kernel of L areeasy to compute.

Application of Forman’s theorem

Now we proceed by using Forman’s method to calculate Det(LA). For thiswe have to define the remaining objects. In particular, C is given by C =f ∈ V |f(M) = 0. To define the volume forms we introduce the notationfZ,k := fZ1,k ∧ · · · ∧ fZN ,k, and similar for fZ,k. Then the volume forms are

vol(A) = fZ,1 ∧ · · · ∧ fZ,M ,

vol(A⊥) = fZ,1 ∧ fZ,M ,

vol(C) = fZ,1 ∧ · · · ∧ fZ,M−1 ,

vol(C⊥) = fZ,M ∧ fZ,M ,

vol(V ) = vol(A⊥) ∧ vol(A) = vol(C) ∧ vol(C⊥) .

Note that we have Image(L) = A and hence Det(πA : Image(L) → A) = 1.Our choice of C renders the matrix representation of L : C → Im(L) lowertriangular. The matrix is given by

1B′M−1 D′M−1

−D′M−1 −A′M−2 1. . .−D′2 −A′1 1

−1 B′1 D′0

,


and Det(L : C → Im(L)) is simply the product of the determinants of theblockmatrices on the diagonal:

limM→∞

M−1∏k=0

det(D′k) = limM→∞

M−1∏k=0

det(δαβ −

2βNiM

∂Zβ

(gαγ∂ZγH

))

= limM→∞

M−1∏k=0

(1− β

M

2Ni∂Zβ

(gβγ∂ZγH

))

= exp(− 2Ni

∫ β

0dτ ∂Zβ

(gβγ∂ZγH

)).

This is already part of the Solari-Kochetov phase, the other part comes fromthe determinants of the projection operators.

It remains to calculate det(πA⊥ρCK : C⊥ → A⊥). For that we assumethat V = C ⊕K holds. To compute the determinant, which is defined withrespect to the given volume forms, we have to calculate the images of thebasis vectors of C⊥ under πA⊥ ρCK . For f ∈ C⊥ and f := ρCK(f) ∈ K itfollows that

f(M) = f(M) . (2.12)

Thus ρCK can be seen as the identification of an element f in the kernel of Lwith its initial condition f(M).

The image of fZα,M is given by

kZα := kZαZβ (1)fZβ ,1 + kZαZβ

(M)fZβ ,M

:= πA⊥ ρCK(fZα,M )

= kZαZβ (1)fZβ ,1 ,

where we have used that we have by definition of ρCK that

kZα(M) = fZα,M (M) = eα

⇒ kZαZβ

(M) = 0 .

And similarly the image of fZα,M is given by

kZα := kZαZβ (1)fZβ ,1 + kZαZβ

(M)fZβ ,M

:= πA⊥ ρCK(fZα,M )

= kZαZβ (1)fZβ ,1 + fZα,M .


Now the determinant can be read off from

det(πA⊥ ρCK : C⊥ → A⊥)vol(A⊥)

= (πA⊥ ρCK : C⊥ → A⊥)∗vol(C⊥)

=∏α

kZαZβ (1)fZβ ,1 ∧∏α

(kZαZβ (1)fZβ ,1 + fZα,M )

= det(kZαZβ (1))vol(A⊥) .

We now put everything together to obtain

limM→∞

DetLA = det(kZαZβ (0)) exp(− 2Ni

∫ β

0dτ ∂Zβ

(gβγ∂ZγH

)), (2.13)

where kZα(0) = limM→∞ kZα(1). The strategy of the next three subsections

will be to connect kZαZβ (0) to the continuous action. To that end we firstcalculate the continuum limit of L.

Continuum limit of L

To obtain the continuum limit of L we have to calculate the matrix entriesup to first order in β/M . We begin with a rather detailed calculation forAk:

−(g−1k Ak)αβ = gαγk Ak,γβ ≈

2βNiM

∂Zβ(gαγ∂ZγH

),

where we used that

lnFk+1,k

Fk,k≈ ∂Zk,δ lnFk,k ∆Zk+1,δ

and

gαγ∂Zγ∂Zβ∂Zδ lnF ∂τ Zδ = − 2Ni

(∂Zβg

αγ)∂ZγH .

Similar calculations yield:

(g−1k Bk)αβ ≈ −

2βNiM

∂Zβ

(gαγ∂ZγH

),

(g−1k Dk)αβ ≈ δαβ −

2βNiM

∂Zβ

(gαγ∂ZγH

),

(g−1k−1Dk)αβ ≈ δαβ −

2βNiM

∂Zβ(gαγ∂ZγH

).

Up to a factor −β/M this leads to the first order differential operator

L :=

(2Ni∂Zγ (gαβ∂ZβH) δγα∂τ + 2

Ni∂Zγ (gαβ∂ZβH)

δγβ∂τ − 2

Ni∂Zγ (gαβ∂ZαH) − 2

Ni∂Zγ (gαβ∂ZαH)

).


To make the connection with the continuous action (2.8) we consider its firstvariation, including variation of the boundary values. This leads to

δS =Ni

2

[∂Zβ lnF (β)δZβ + ∂Zα lnF (0)δZα(0)

]+Ni

2

∫ β

0dτδZα

(∂Zα∂Zβ lnF ∂τ Zβ − ∂ZαH

)− Ni

2

∫ β

0dτδZβ

(∂Zβ∂Zα lnF ∂τZα + ∂ZβH

). (2.14)

The equations of motion are

∂τ Zγ =2Nigαγ∂ZαH

∂τZγ = − 2Nigγβ∂ZβH .

In particular, the linearised equations of motion are given by

L

(δZγδZγ

)= 0 .

Next we view kZαZβ as an element of the kernel of L. This allows us to relate

kZαZβ (0) to solutions of the classical equations of motion and in particular tothe action of these solutions.

Relating kZαZβ (0) to classical solutions

The kernel of L is spanned by ∂Zα(0)(Zγ(t)eγ+Zγ(t)eγ) and ∂Zα(β)(Zγ(t)eγ+Zγ(t)eγ), where Z(t) and Z(t) are solutions of the classical equations ofmotion, which are seen as functions of their initial values Z(0) and Z(β).

We are interested in the special element that fulfils kZα(β) = eα. It canbe written as (no summation convention)

kZα(t) =∑γ

(∂Zα(0)Zγ(β))−1∂Zα(0)(Zγ(t)eγ + Zγ(t)eγ) .

In the last step we express det(kZαZβ (0)) by second derivatives of the action.Therefore consider

∂Zα(0)∂Zβ(β)2S/Ni = ∂Zα(0)

(∂Zβ lnF

∣∣Z=Z(β),Z=Z(β)

)=(∂Zγ∂Zβ lnF

) ∣∣Z=Z(β),Z=Z(β)

∂Zα(0)Zγ(β)

= gγβ(β) ∂Zα(0)Zγ(β)

⇒ ∂Zα(0)Zδ(β) = gδβ(β)∂Zα(0)∂Zβ(β)2S/Ni ,


where we used (2.14). This leads to

kZαZδ (0) = (∂Zα(0)Zδ(β))−1 =(gδβ(β)∂Zα(0)∂Zβ(β)2S/Ni

)−1.

Hence we have

limM→∞

detLA = det(gδβ(β)∂Zα(0)∂Zβ(β)2S/Ni

)−1

× exp(− 2Ni

∫ β

0dτ ∂Zβ

(gβγ∂ZγH

)).

After some additional manipulations we obtain the symmetrised version ofthe inverse square root of the determinant

limM→∞

det−1/2(LA) =[det(∂Z(0)∂Z(β)2S/Ni)

g1/2(β)g1/2(0)

]1/2

× exp(

12Ni

∫ β

0dτ [∂Zβ (gβγ∂ZγH) + ∂Zβ (gγβ∂ZγH)]

), (2.15)

which is precisely the statement (2.1).

Conclusion

In the first chapter we proved a general version of the Pruisken-Schaferhyperbolic Hubbard-Stratonovich transformation. Previous results concern-ing pseudounitary and pseudoorthogonal symmetry are obtained as specialcases. The method of the proof also shows why the transformation holds:The Pruisken-Schafer domain can be seen as a deformation of the standardGaussian domain of integration. Thus the transformation is valid, since theintegrand is holomorphic and the domain of integration can be deformedwithout changing the value of the integral. Deformation of the domain ofintegration requires integration of chains against forms. In this setting, thealternating sign factors appearing in the pseudoorthogonal case have a nat-ural explanation. More precisely, we find that the alternating sign factorsare induced by the sign of the product of all weights in Σ+(p⊕Q−, h).

Apart from the Pruisken-Schafer transformation we also considered thewell-established Schafer-Wegner transformation. We showed that the latteris also a deformation of the standard Gaussian domain.

To sum up, the first chapter yields a unified view on the different Hubbard-Stratonovich identities, and our result allows to rigorously apply the Pruisken-Schafer transformation. Using these transformation implies in particular thepossibility of obtaining results beyond the large N limit.

In the second chapter we developed a path integral representation ofthe grand canonical partition function for an interacting granular fermionicsystem in terms of bosonic variables. In particular, we have discussed twodifferent derivations of this representation. The first derivation uses thecolour-flavour transformation and the time discrete Grassmann path in-tegral, whereas the second derivation illuminates the structure of the un-derlying Hilbert spaces. Furthermore we have computed contributions ofquadratic fluctuations to the large N limit in terms of the correspondingclassical system. The result, concerning the contributions of fluctuations,applies to a wide class of generalised coherent state path integrals. An in-teresting prospect for the future is to establish applications of the novelbosonisation method developed in chapter two.

57

58 CHAPTER 2. CONCLUSION

Appendix A

Techniques needed inchapter one

A.1 Basic constructions and useful relations

A.1.1 Setting

To make the appendix self contained we repeat the setting of theorem 1.2.For any hermitian s ∈ gl(n,C), with s2 = 1, we have two involutions θ(X) =sXs−1 and γ(X) = −sX†s−1 on gl(n,C). Let τi be additional involutionson gl(n,C), which have to commute with each other and with θ and γ. Thisleads to the definition

Q := Q ∈ gl(n,C)|Q = −γ(Q) and ∀i : Q = σiτi(Q) ,

where σi ∈ ±1, and the τi have to be such that s ∈ Q.

A.1.2 Symmetries of Q

Define a Lie algebra

g := X ∈ gl(n,C)|X = γ(X) and ∀i : X = τi(X) ,

and let τ be either τi or γ. For X ∈ g and X ′ ∈ Q, the basic calculation

τ([X,X ′]) = [τ(X), τ(X ′)] = [X, τ(X ′)]

shows that g is the Lie algebra of the symmetry group of Q.

A.1.3 Commutation relations

The decomposition of Q into the plus and minus one eigenspaces of θ givesa decomposition into the hermitian and antihermitian parts denoted by Q+

and Q−. Similarly, θ gives the decomposition g = k⊕ p, where k is the plus

59

60 APPENDIX A. TECHNIQUES NEEDED IN CHAPTER ONE

one eigenspace and p the minus one eigenspace. This leads to the followingcommmutation relations:

[k, k] ⊂ k , [k, p] ⊂ p , [p, p] ⊂ k , [Q+,Q−] ⊂ p ,[Q±,Q±] ⊂ k , [k,Q±] ⊂ Q± , [p,Q±] ⊂ Q∓ .

(A.1)

Defining

f : p⊕Q− → p⊕Q−

X 7→ 12

[X, s] ,

it is easy to check that the inverse is given by

f−1 : p⊕Q− → p⊕Q−

X 7→ 12

[X, s−1] .

In particular we have

f(p) = Q− and f(Q−) = p . (A.2)

Note that we always have s ∈ Q+.

A.1.4 Decompositions

For H,X, Y ∈ gl(n,C) with H = H† we have

Tr([H,X]Y †) = Tr(X[Y †, H]) = Tr(X([H,Y ])†) .

This shows that [H, ·] is hermitian with respect to Tr(XY †), which is ahermitian scalar product on the space of complex matrices. Hence [H, ·] hasreal eigenvalues, and the corresponding eigenspaces are orthogonal. In thefollowing we choose sets of commuting hermitian matrices, and diagonalisethe commutator or adjoint action simultaneously.

Decomposition of g

The first set we consider is a maximal Abelian subalgebra a of p. We considerthe adjoint action of a on g. This leads to the decomposition

g = g0 ⊕⊕

α∈Σ(g,a)

gα .

In this setting the eigenvalues α are called roots. The set of roots is denotedby Σ(g, a). For Z ∈ gα and H ∈ a we have

[H, θ(Z)] = −θ([H,Z]) = −α(H)θ(Z) ,

A.1. BASIC CONSTRUCTIONS AND USEFUL RELATIONS 61

and hence θ(gα) = g−α. This shows that

g±,α := Fix±θ(gα ⊕ g−α) (A.3)

is well defined. Choosing Ho ∈ a with α(Ho) 6= 0 for all α, we have a notionof positivity for the roots α. Let Σ+(g, a) denote the set of positive roots.Then we have the decomposition

g = g0 ⊕⊕

α∈Σ+(g,a)

(gα ⊕ g−α) . (A.4)

Note that g0 = ZK(a)⊕ a. Furthermore we can also define a mapping φ as

φ : gα ⊕ g−α → gα ⊕ g−α

Z + Z ′ 7→ 1α(Ho)

[Ho, Z + Z ′] = Z − Z ′ , (A.5)

which has the properties

φ φ = id and φ(g±,α) = g∓,α . (A.6)

The mapping φ obviously extends to a mapping on⊕

α∈Σ+(g,a) (gα ⊕ g−α).Furthermore for X ∈ g±,α and H ∈ a we have

[H,X] = α(H)φ(X) and [H,φ(X)] = α(H)X . (A.7)

In addition we define orthogonal projections

π±,α : g→ g±,α . (A.8)

Next we discuss additional decompositions. For each decomposition we de-fine the appropriate mappings φ and π±,α. Although they are also called φand π±,α, it is always clear from the context which mapping is meant.

Decomposition of Q

Next, we consider the adjoint action of a on Q, which gives us the decom-position

Q = Q0 ⊕⊕

α∈Σ+(Q,a)

(Qα ⊕Q−α) . (A.9)

Using the same reasoning as above we obtain that

Q±,α := Fix±θ(Qα ⊕Q−α) ⊂ Q± (A.10)


is well defined. The same is true for Q±,0 := Fix±θ(Q0) ⊂ Q±. ChoosingHo ∈ a with α(Ho) 6= 0 for all α ∈ Σ(Q, a), we again have a notion ofpositivity. Hence we have the decompositions

Q± = Q±,0 ⊕⊕

α∈Σ+(Q,a)

Q±,α . (A.11)

Now we can define

φ : Qα ⊕Q−α → Qα ⊕Q−α

Z + Z ′ 7→ 1α(Ho)

[Ho, Z + Z ′] = Z − Z ′ . (A.12)

Note that φ satisfies the corresponding relations to (A.6) and (A.7). Inaddition we define orthogonal projections πα:

π±,α : Q → Q±,α . (A.13)

Decomposition of k⊕Q+

Secondly, we choose a maximal Abelian subalgebra h of Q+ such that s ∈ h

and consider its adjoint action on

g := k⊕Q+ . (A.14)

Similar to the above we obtain the decomposition

g = Zk(h)⊕ h⊕⊕

α∈Σ+(g,h)

(gα ⊕ g−α) . (A.15)

In addition we define the mapping φ as

φ : gα ⊕ g−α → gα ⊕ g−α

Z + Z ′ 7→ 1α(Ho)

[Ho, Z + Z ′] = Z − Z ′ . (A.16)

Note that H0 is of course choosen such that α(H0) 6= 0 for all α ∈ Σ(g, h)and φ satisfies the analogous relations to (A.6) and (A.7). Defining

g±,α := Fix±γ(gα ⊕ g−α) (A.17)

the orthogonal projection operators are denoted by

π±,α : g→ g±,α . (A.18)

A.1. BASIC CONSTRUCTIONS AND USEFUL RELATIONS 63

Decomposition of p⊕Q−

Next we consider the adjoint action of h on

Q := p⊕Q− . (A.19)

In the following we list the analogous definitions and objects: The decom-position of Q is given by

Q =⊕

α∈Σ+(Q,h)

(Qα ⊕ Q−α) . (A.20)

Note that, using s ∈ h and (A.2), it can be easily checked that Q0 is trivial.Since we have α(s) 6= 0 for all α it is useful to use s to obtain a notion ofpositivity for the weights. This special choice implies in particular that theproperty α > 0 is invariant under the action of the Weyl group, since s isAd(K) invariant. Furthermore we define

Q±,α := Fix±γ(Qα ⊕ Q−α) , (A.21)

and

φ : Qα ⊕ Q−α → Qα ⊕ Q−α

Z + Z ′ 7→ 1α(s)

[s, Z + Z ′] = Z − Z ′ , (A.22)

which satisfies the corresponding relations to (A.6) and (A.7). In additionwe define

Q±,α := Fix±γ(Qα ⊕ Q−α) (A.23)

and the orthogonal projections

π±,α : Q → Q±,α . (A.24)

A.1.5 Orthogonality relations

We discuss orthogonality with respect to Tr(XY †). In particular we areinterested in the corresponding relations for B(X,Y ) = Tr(XY ). A typicalargument for X ∈ k and Y ∈ p goes as follows:

B(X,Y ) = Tr(XY †) = Tr(sXssY †s) = −Tr(XY †) = 0 . (A.25)

The same holds for X ∈ Q+⊕Y ∈ Q−. This implies that the decompositionsg = k ⊕ p and Q = Q+ ⊕ Q− are orthogonal. The following relations forXα ∈ Q+,α, X,X ′ ∈

⊕α>0Q+α hold:

a) B(Xα, Xβ) = −B(φ(Xα), φ(Xβ)) = δα,βB(Xα, Xα)


b) B(X,X ′) = −B(φ(X), φ(X ′))

c) B(Xα ± φ(Xα), Xβ ± φ(Xβ)) = 0

d) B(Xα + φ(Xα), Xβ − φ(Xβ)) = δα,β2B(Xα, Xα) .

Indeed, note that Xα ± φ(Xα) ∈ Q±α, and that for X,Y ∈ Q Tr(XY †) =Tr(Xθ(Y )) holds. This immediately gives c). Using (A.25), gives a) andhence d). b) is an immediate consequence of a).

A.2 Three additional arguments

A.2.1 Reparametrisation RI

Here we want to show thatRI is injective and regular. Note thatNK(a)/ZK(a)is the Weyl group, which acts simply transitive on the Weyl chambers [14].Let H ∈ ao+ and k ∈ K, then we have

kHk−1 = k′H ′k′−1

⇔ k′−1kH(k′−1k)−1 = H ′ .

This shows that k := k′−1k ∈ NK(a) and thus k = wz with w in theWeyl group and z ∈ ZK(a). It follows that wHw−1 = H ′, which impliesthat w = 1, H = H ′, and hence [k] = [1]. This proves injectivity. Thedifferential is given by

d(kHk−1) = Ad(k)([k−1dk,H] + dH

)= Ad(k)

∑α∈Σ+(g,a)

α(H) π−,α φ(k−1dk) + dH

.

All α(H) 6= 0, and hence the differential has full rank.

A.2.2 Pullback of dQ

In this subsection we use

PS R : G/ZK(a)× ao+ × h′ → Q([g], H,H ′) 7→ g(H +H ′)g−1 .

to pullback the constant volume form dQ on Q. In the following we use thenotation and the constructions of section A.1.4. Consider the adjoint actionof h on

g⊕Q = g⊕ Q ,

A.2. THREE ADDITIONAL ARGUMENTS 65

which leads to the decomposition

Q = h⊕⊕

α∈Σ+(g,h)

g−,α ⊕⊕

α∈Σ+(Q,h)

Q−,α . (A.26)

The corresponding set of weights is given by

Σ+ := Σ+(g, h) ∪ Σ+(Q, h) .

For α ∈ Σ+ we define a basis eα,i of g−,α and Q−,α. Note that i ∈1, . . . , dα, where dα denotes the dimension of the corresponding weightspace of α. Furthermore let dH denote a constant volume form on h. Nowwe can compute (PS R)∗dQ:

(PS R)∗dQ

= dQ[Ad(g)([g−1dg, H +H ′] + dH + dH ′)]

= dQ[∑α∈Σ+

α(H +H ′)φ π+,α(g−1dg) + dH + dH ′]

=∏α∈Σ+

α(H +H ′)dα∧α∈Σ

dα∧i=1

[deα,i φ π+,α(g−1dg)

]∧ dH

=: ∆(H +H ′) dµ([g]) ∧ dH . (A.27)

The first equality in (A.27) is obtained by elementary calculation. For thesecond equality we have used equations (A.20), (A.15) and (A.26) and thefact that det(exp([X, ·])) = 1 for X ∈ g. The third equality just uses thebasis eα,i we defined above. Note that comparison of the second and thelast line in (A.27) shows that dµ([g]) is a left invariant volume form onG/ZK(a).

Note that α(H +H ′) ∈ R, since [H +H ′, ·] is hermitian with respect toTr(XY †). For α ∈ Σk⊕Q+ we have α(H ′) = 0. Therefore we obtain:

∆(H +H ′) =∏

α∈Σp⊕Q− ,α>0

α(H +H ′)dα ·∏

α∈Σk⊕Q+,α>0

α(H)dα .

A.2.3 Contributions from ∂a+

Here we give the detailed argument showing the contributions from ∂a+ areirrelevant for PS, ePS and DePS, since they are at least of codimensiontwo. In doing so we recall the argument for the well definedness of theseparametrisations.

Let Hi ∈ a+,c ∩ ∂a+ be an edge of a+. We show that the restrictionof PSc to the domain of definition with hi = 0 yields a domain of at leastcodimension two. It is well known that the dimension of the isotropy group


of a changes at the boundary of a+. In particular dimZK(∑

j 6=i hjHj) >

dimZK(a). This can be seen from the fact that to each face of the boundaryof the Weyl chamber there is associated a restricted root α with root spacegα and α(

∑j 6=i h

jHj) = 0. The group generated by Fixθ(gα ⊕ g−α) leavesthe face invariant, and Fixθ(gα ⊕ g−α) 6⊂ Zk(a). If we restrict PS RI tosuch a face of a+, we can replace K/ZK(a) by the lower dimensional spaceK/ZK,i, with ZK,i := ZK(

∑j 6=i h

jHj) without changing the image of theparametrisation. In fact the eigenspace decomposition of Q with respect to a

restricts to an eigenspace decomposition with respect to the smaller algebra∑

j 6=i hjHj. Hence all parametrisations restrict to parametrisations where

the quotient group is ZK,i. This completes the argument for PSc.For ePS we have to argue that the analogous restriction is well defined.

For that it is enough to note that for X ∈ Lie(Zk,i) we have [X,Ej ] = 0if j 6= i. The analogous restriction of DePS is well defined for the samereason. Thus we see that for both ePS and DePS the contributions from∂a+ are of codimension at least two.

Appendix B

Techniques needed inchapter two

A major tool of chapter two are generalised coherent states. These coher-ent states are built using group representations on fermionic Fock space.All groups representations we consider are subgroups of the large group ofcanonical transformations of fermionic Fock space. A second ingredient is atheorem by Howe on dual groups within the group of canonical transforma-tions. This theorem leads us to the colour-flavour transformation.

Everything stated in this appendix is well known. Coherent states arediscussed extensively in Perelomov [25]. Howe pairs are discussed in [27, 28].The colour-flavour transformation is discussed in [29, 30]. Since the materialis spread over different articles, and to introduce the notation we are usingin chapter two, we develop the necessary constructions explicitly. However,some mathematical theorems are stated without proof.

B.1 Canonical transformations of fermionic Fockspace

Our first aim is to construct group actions on fermionic Fock space. Denotethe creation operators and annihilation operators by c†n and cn. Then we canuse the anticommutator to define a bilinear form by 〈·, ·〉 := ·, ·. Since c†nand cn satisfy canonical anticommutation relations (CAR), the basis definedby

vi :=1√2

(ci + c†i )

vN+i :=i√2

(ci − c†i )

67

68 APPENDIX B. TECHNIQUES NEEDED IN CHAPTER TWO

is orthonormal with respect to 〈·, ·〉. It is easy to check that

〈[vivj , vk], vl〉 = −〈vk, [vivj , vl]〉

holds. Therefore [vivj , ·] generates infinitesimal isometries of 〈·, ·〉 and thuscanonical transformations. For an antisymmetric matrix Z with complexentries the following identities may also easily be checked:

Zij [vivj , vk] = 2viZik ,

[viZijvj , vkZklvl] = vi

([Z, Z]

)ijvj .

The last line gives a Lie-algebra isomorphism from so(2N,C) to the vec-torspace generated by vivj with the commutator as a Lie-bracket and thealgebraic structure from the anticommutator ·, ·. Next we transform tothe natural basis given directly by creation and annihilation operators. Inorder to do this we have to split up Z into N ×N blocks

Z =(

A′ B′

−B′t D′

),

where A′ and D′ are antisymmetric and B′ is an arbitrary complex matrix.Then we have

vtZv =12

(c†c)(A′ +D′ + iB′ + iB′t A′ −D′ − iB′ + iB′t

A′ −D′ + iB′ − iB′t A′ +D′ − iB′ − iB′t)

( cc† )

=:12

(c†c)(A BC −At

)( cc† ) =:

12

(c†c)M ( cc† ) =: M,

where A is an arbitrary complex matrix, whereas B and C are antisymmetriccomplex matrices. This is equivalent to the condition M = −ΣxM

tΣx, withΣx := σx ⊗ 1N . The set of M matrices is a Lie algebra isomorphic toso(2N,C) and the mappingˆ: M 7→ M is a Lie algebra isomorphism.

The commutator action of M on the creation and annihilation operatorsis given by

[M, (c†c)] = (c†c)(A BC −At

)and thus we have

exp(M)(c†c) exp(−M) = exp([M, ·])(c†c) = (c†c) exp(M) .

Demanding that the canonical transformations respect (c†)† = c impliesthat M = −M †. Hence Z has to be real and the corresponding Lie algebrais so(N,R). Exponentiating the Lie algebra representation M leads to aunitary representation of the spin group Spin(N), which is a double coveringof the special orthogonal group.

B.1. CANONICAL TRANSF. OF FERMIONIC FOCK SPACE 69

B.1.1 Matrix elements

Denote by 〈ψ| and |ψ〉 standard fermionic coherent states. In the followingwe calculate the matrix element 〈ψ| exp(M)|ψ〉. To proceed we decompose

g :=(a bc d

):= exp(M)

and then lift the decomposition to a decomposition of g := exp(M). If D isinvertible the Gaussian decomposition of g is given by

g =(1 bd−1

0 1

)(a− bd−1c 0

0 d

)(1 0

d−1c 1

)=: g1 g2 g3 .

The single factors however live in the complex orthogonal group defined byg ∈ GL(N,C)|gtΣxg = Σx. Defining

M1 :=(

0 bd−1

0 0

)and M2 :=

(0 0

d−1c 0

)we have g1 = exp(M1) and g3 = exp(M3). Hence, defining g1 := exp(M1),g3 := exp(M3) and g2 := g−1

1 gg−13 , we obtain the decomposition g = g1g2g3.

Note that π(g2) = g2, where π denotes the covering homomorphism. As-suming that there exists a matrix M2 such that g2 = σ exp M2, with σ beingeither plus or minus one, it is easy to show that

〈ψ| exp(M)|ψ〉 = 〈ψ|g1g2g3|ψ〉

= σDet1/2(d) e12ψbd−1ψe

12ψd−1cψe−ψd

−1ψ (B.1)

holds. In addition we have

〈ψ| exp(M)|0〉 = σDet1/2(d)e12ψbd−1ψ .

Using that g−1 = g† we obtain

〈0| exp(−M)|ψ〉 = σDet1/2(d†)e12ψ(bd−1)†ψ ,

and dd† = (1 + (bd−1)†bd−1)−1. Defining Z := bd−1 we have

〈ψ| exp(M)|0〉〈0| exp(−M)|ψ〉 = Det1/2(1 + Z†Z)e12ψZψe

12ψZ†ψ . (B.2)

It is important to note that the matrix elements (B.1) and (B.2) only dependon g and not on g. Furthermore, there is no non-analyticity hidden in Det1/2

since Z is antisymmetric and

Det1/2(1 + Z†Z) = Pf(Z −11 Z†

).

Pf denotes the Pfaffian.


B.2 Fermionic Howe pairs and colour-flavour trans-formation

A fermionic Howe dual pair (G,K) is a pair of two subgroups G and K ofSO(2N) that centralise each other. In the bosonic case it becomes importantto demand in addition that the pair is reductive. The mathematical theoryof these pairs is developed in [28] and physical applications are given in [27].We formulate a basic theorem on dual pairs in our context.

Theorem B.1. Let (G,K) be a fermionic Howe dual pair and H be theHamiltonian of a granular fermionic system, i.e., H is a polynomial in cre-ation and annihilation operators and has a classical symmetry group K.Then H is a polynomial in the generators of G. In particular the quadraticinvariants of K are the generators of G.

The fermionic pairs are listed in table B.1. H denotes the subgroup of G

SO(2N) K G H

N = (p+ q)Ni U(Ni) U(p+ q) U(p)×U(q)N = NeNi O(Ni) SO(2Ne) U(Ne)N = 2NeNi USp(2Ni) USp(2Ne) U(Ne)

Table B.1: Fermionic Howe pairs

which leaves the vacuum state |0〉 invariant. It is known from [27, 28] thatthere are two different representations for the projector onto the subspaceof states which are invariant under the K action, i.e. k|v〉 = ρ(k)|v〉 fork ∈ K. Here ρ is a one dimensional representation of G. The differentrepresentations for the projector P are

P =∫Kdk ρ(k−1) k =

∫Gdg g|0〉〈0|g−1 .

In the following we will evaluate 〈ψ|P |ψ〉 for the different representationsfor each of the three fermionic Howe dual pairs. Note that we will extendthe notation of the previous section appropriately. Lie algebra elements ofLie(G) are denoted by Me and elements of Lie(K) are denoted by Mi andso on. Note that Mi/e inherits symmetry properties of M ∈ so(2N), whichwe do not state explicitely in the following. Only the additional symmetriesare discussed. Eventually an additional two by two substructure requiresintroducing a ± index.

B.2. FERMIONIC HOWE PAIRS AND CFT 71

B.2.1 U(p + q)⊗ U(Ni)

In the following we state how to realise u(p + q) and u(Ni) as commutingsubalgebras of so(2(p+ q)Ni). u(p+ q) is given by matrices of the form

Me :=

Ae+ 0 0 Be+

0 Ae− −Bte+ 0

0 Be+ −Ate+ 0−B†e+ 0 0 −Ate−

⊗ 1Ni ,where Ae+ is a p× p matrix and Ae− is a q × q matrix. u(N) is realised by

Mi :=

1p ⊗Ai 0 0 0

0 −1q ⊗Ati 0 00 0 −1p ⊗Ati 00 0 0 1q ⊗Ai

.

There are no additional symmetry requirements beyond the ones the matri-ces inherit from so(2(p+ q)N). To evaluate the matrix elements (B.1) and(B.2), define u := exp(Ai) and

g :=(ae bece de

):= exp

(Ae+ Be+−B†e+ −Ate−

).

Next we calculate exp(Me). To this end it is useful that

exp(Ae− −Bt

e+

Be+ −Ate+

)=(

0 1

1 0

)k

(0 1

1 0

)holds. Then we can read off

exp(Me) =

ae 0 0 be0 de ce 00 be ae 0ce 0 0 de

⊗ 1Ni .Hence we have

Z = Ze ⊗ 1Ni =(

0 bed−1e

cea−1e 0

)⊗ 1Ni .

In particular Z is antisymmetric, which implies that the right hand side onlydepends on Z+ := bed

−1e . Thus we have∫

U(Ni)du eψuψ =

∫U(p+q)

dk DetNi/2(1 + Z†eZe) e12ψZψe

12ψZ†ψ

=∫

U(p+q)dk DetNi(1 + Z†+Z+) eψ+Z+⊗1Ni ψ−eψ−Z

†+⊗1Niψ+ , (B.3)


where

u =(1p ⊗ u 0

0 1q ⊗ u

).

In fact the set of Z+ matrices is a parametrisation of the quotient spaceU(p+ q)/U(p)×U(q). However for our purpose it is not necessary to knowthe exact form of the induced measure dµ(Z†, Z).

B.2.2 SO(2Ne)×O(Ni)

In the following we state how to realise so(2Ne) and o(Ni) as commutingsubalgebras of so(2NeNi). so(2Ne) is given by matrices

Me :=(Ae Be−Be −Ate

)⊗ 1Ni

and o(Ni) is given by

Mi := 1Ni ⊗(Ai 00 −Ati

),

where the entries of Ai have to be real. Defining o := exp(Ai) and

O :=(ae bece de

):= exp

(Ae Be−Be −Ate

)the formulas for the matrix elements (B.1) and (B.2) yield∫

O(Ni)do eψ1Ne⊗oψ =

∫SO(2Ne)

dO DetNi/2(1 + Z†eZe)e12ψZψe

12ψZ†ψ , (B.4)

where Z = Ze ⊗ 1Ni and Ze = bed−1e = −Zte parametrises the coset space

SO(2Ne)/U(Ne).

B.2.3 USp(2Ni)× USp(2Ne)

In the following we state how to realise usp(2Ni) and usp(2Ne) as commutingsubalgebras of so(4NiNe). usp(2Ni) is given by

Mi := 1Ne ⊗

Ai Bi 0 0−Bi −Ati 0 0

0 0 −Ati Bi0 0 −Bi Ai

,

where Bi is symmetric. usp(2Ne) is given by

Me :=

Ae 0 0 Be0 Ae −Be 00 Be −Ate 0−Be 0 0 −Ate

⊗ 1Ni ,


where Be is symmetric. A short calculation shows that these two subalgebrasindeed commute. Defining

s = exp(Ai Bi−Bi −Ati

)and S =

(ae bece de

)= exp

(Ae BeCe −Ate

)and noting that

exp(Ae −BeBe −Ate

)=(

0 1

1 0

)S

(0 1

1 0

),

we can read off

exp(Me) =

ae 0 0 be0 de ce 00 be ae 0ce 0 0 de

⊗ 1Ni .Hence we have

Z = Ze ⊗ 1Ni =(

0 bed−1e

cea−1e 0

)⊗ 1Ni .

It can be easily checked that JZeJ t = Ze and sZes = −Ze. The formulasfor the matrix elements (B.1) and (B.2) yield∫

USp(2Ni)ds eψ(1Ne⊗s)ψ =

∫USp(2Ne)

dS DetNi/2(1 + Z†eZe) e12ψZψe

12ψZ†ψ .

(B.5)

B.2.4 Invariant measure on G/H

In this subsection we want to simplify the right hand side of the colour-flavour transformation. To this end we want to use the following version ofFubini’s theorem: ∫

Gdgf(g) =

∫G/H

d(gH)∫Hdhf(gh) . (B.6)

Here, d(gH) denotes the invariant measure on the coset space G/H, whereH is a (closed and connected) subgroup of G. Since we are here dealingwith a simpler situation than in the last section, we simplify our notationaccordingly. For an element g ∈ G we write

g =(a bc d

),

and we define Z := bd−1. We will show that Z parametrises the coset spaceG/H up to a set of measure zero and construct a left invariant measure


dµ(Z†, Z). Here, Z differs from the matrix defined in the previous section.Note that left invariance and normalisation uniquely determine the measured(gH). In our case the H integration is trivial, since the integrand onlydepends on Z. In the following we treat the three different fermionic Howepairs simultaneously. We define s = Diag(1k,−1l). For the first Howe pairset k = p and l = q and for the other two set k = l = Ne.

Symmetry class V1 V2

unitary Wp,q Wq,p

orthogonal Z ∈WNe,Ne |Zt = −Z Z ∈WNe,Ne |Zt = −Zunitary symplectic Z ∈WNe,Ne |Zt = Z Z ∈WNe,Ne |Zt = Z

Table B.2: Wk,l := Hom(Cl,Ck) is used. The different symmetries of Z =bd−1 and the corresponding vectorspace for each of the three settings isdefined.

Then it is clear that Ad(G)s is isomorphic to G/H. Note that for g ∈ Gwe have

Ad(g)s ≡(a bc d

)s

(a bc d

)−1

=(

1 bd−1

ca−1 1

)s

(1 bd−1

ca−1 1

)−1

.

It can be checked that bd−1 = −(ca−1)†. Hence Z = bd−1 parametrisesAd(G)s and thus G/H. For the construction of the left invariant measureon G/H we have to consider the adjoint action of G on Ad(G)s. Here it isconvenient to consider first the adjoint action on

f(Z, Z) :=(

1 Z

Z 1

)s

(1 Z

Z 1

)−1

with Z ∈ V1 and Z ∈ V2. It induces an action on V1 and V2. We have

Ad(g)f(Z, Z) = f(ρ1g(Z), ρ2

g(Z)) ,

where ρ1 and ρ2 are given by

ρ1g(Z) := (aZ + b)(cZ + d)−1

ρ2g(Z) := −(c− dZ)(a− bZ)−1 .

Note that ρ1 gives the left action on G/H in the parametrisation given by thecoordinates Z = bd−1. Moreover we can define a bilinear form Tr(df ⊗ df)1,

1It is convenient to show invariance of the unsymmetrised bilinear form and to postpone(anti)symmetrisation.


where d denote the outer derivative. It is invariant since

ρ∗g Tr(df ⊗ df) = Tr(d(f ρg)⊗ d(f ρg)) = Tr(Ad(g)[df ⊗ df ])

= Tr(df ⊗ df) .

Defining v := (1 − ZZ)−1 and w = (1 − ZZ)−1 a simple calculation showsthat

Tr(df ⊗ df)/4 = −Tr(wdZ ⊗ vdZ)− Tr(vdZ ⊗ wdZ) .

Since s commutes with the differential of ρg, we can define an invariant twoform as

ω(Z,Z)(X,Y ) :=14

Tr(df ⊗ df)(Z,Z)(sX, Y )

= Tr(vdZ ⊗ wdZ)− Tr(wdZ ⊗ vZ)(X,Y )

= Tr(vdZ ∧ wdZ)(X,Y ) .

Next we restrict ourselves to the subspace Z = −Z†. This is compatible withthe group action since ρ2

g(−Z†) = −[ρ1g(Z)]†. Hence we have an invariant

(symplectic) two form given by:

ωZ := ωZ,−Z† |V×V

with V := (Z,−Z†)|Z ∈ V1. The two form ω leads to the (nontrivial)invariant volume form

dµ(Z†, Z) := c ωn = Det−2Ne(1 + Z†Z)∧i,j

dZ†ij∧i,j

dZij , (B.7)

where c is a normalisation constant and n = dimCV1.

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ErklarungIch versichere, dass ich die von mir vorgelegte Dissertation selbstandig ange-fertigt, die benutzten Quellen und Hilfsmittel vollstandig angegeben und dieStellen der Arbeit – einschließlich Tabellen, Karten und Abbildungen –, dieanderen Werken im Wortlaut oder dem Sinn nach entnommen sind, in je-dem Einzelfall als Entlehnung kenntlich gemacht habe; dass diese Disserta-tion noch keiner anderen Fakultat oder Universitat zur Prufung vorgelegenhat; dass sie noch nicht veroffentlicht worden ist sowie, dass ich eine solcheVeroffentlichung vor Abschluss des Promotionsverfahrens nicht vornehmenwerde.Die Bestimmungen der Promotionsordnung sind mir bekannt. Die von mirvorgelegte Dissertation ist von Herrn Prof. Dr. Martin Zirnbauer betreutworden.

Jakob Muller-Hill

Danksagung

Zunachst bedanke ich mich bei Prof. Zirnbauer fur die interessanten Prob-lemstellungen und die Gesprache. Desweiteren bedanke ich mich bei Prof.Heinzner fur die vielen Diskussionen und sehr konstruktive Kritik am erstenTeil der Arbeit, und bei Prof. Hajdu fur das Interesse und die ermutigendenWorte. Desweiteren haben Thomas Bliem, Stefan Borghoff, Oli Goertsches,Tillmann Jentsch, Rochus Klesse, Tobias Luck, Andrea Wolff und ThomasZell mir durch viele Gesprache und Hinweise sehr geholfen. Hier haben sichnaturlich meine direkten Burokollegen am meisten uber Pruisken-Schaferund “Kegel mit angeklebten Flugeln” anhoren mussen.

Grundsatzlich war die Arbeit nur durch die Unterstutzung meiner FrauEva und seit kurzem auch durch die Friedfertigkeit des kleinen Johannesmoglich.

Die Arbeit wurde von der DFG im Rahmen des SFB/TR 12 und vonder Bonn-Cologne Graduate School of Physics and Astronomy unterstutzt.

hyperbolic hubbard-stratonovich transformations and bosonisation of granular fermionic systems

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